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Nākamais: Literatūra Augstāk: vallievads2ht Previous: 10.2. Funkcijas pārtraukuma punkti un to

11. Pielikums

I.1.Konstruēt funkciju grafikus.

  1. $ f(x)=\vert 1-x\vert-3\vert x+2\vert+x$;
  2. $ f(x)=\vert x-3\vert-2\vert x+1\vert-x+1$;
  3. $ f(x)=3\vert x+1\vert-\vert x-1\vert+6$;
  4. $ f(x)=2\vert x-1\vert-\vert 2+x\vert+x$;
  5. $ f(x)=\vert 2-4x\vert-\vert 4+2x\vert-x$;
  6. $ f(x)=\vert 2-x\vert+\vert 3+x\vert+x$;
  7. $ f(x)=\vert x-5\vert+3\vert x-1\vert-x+1$;
  8. $ f(x)=\vert x-2\vert+\vert x-8\vert-5-x$;
  9. $ f(x)=\vert x-1\vert+\vert x-7\vert-6$;
  10. $ f(x)=\vert 2x-1\vert-\vert x+1\vert+x-2$;
  11. $ f(x)=\vert x+2\vert-\vert x+3\vert+\vert 4-x$;
  12. $ f(x)=\vert x\vert+3x+1-\vert x-6\vert$;
  13. $ f(x)=x+2\vert x+1\vert-\vert 5-x\vert$;
  14. $ f(x)=\vert x-4\vert+\vert x+1\vert-x+3$;
  15. $ f(x)=\vert x-7\vert-\vert 5-x\vert+x+3$;
  16. $ f(x)=\vert 2x-1\vert+\vert x+2\vert-4x+2$;
  17. $ f(x)=\vert 1-3x\vert-\vert 2x+3\vert-4$;
  18. $ f(x)=\vert x-1\vert+\vert 2-x\vert-3+x$;
  19. $ f(x)=\vert x+2\vert-\vert 8-x\vert+2x-2$;
  20. $ f(x)=x-\vert 1-3x\vert+\vert 2x\vert-3$;
  21. $ f(x)=\vert x+1\vert+\vert 2-x\vert-5-x$;
  22. $ f(x)=\vert x+3\vert+2x-\vert 1-x\vert+4$;
  23. $ f(x)=5x-\vert 1-x\vert+\vert 3x+5\vert-4$;
  24. $ f(x)=2\vert x-1\vert-\vert 1-2x\vert+3x+5$;
  25. $ f(x)=x+2-3\vert 5-x\vert+\vert 7+x\vert$.


I.2.Atrisināt vienādojumus.

  1. $ x^2-4x+\vert x-3\vert+3=0$;
  2. $ x^2-6x+\vert x-4\vert+8=0$;
  3. $ x^2+4x+\vert x+1\vert+3=0$;
  4. $ (x+1)^2-2\vert x+1\vert+1=0$;
  5. $ x^2+2x-3\vert x+1\vert+3=0$;
  6. $ x^2+6x+\vert x+2\vert+8=0$;
  7. $ \vert 2x+1\vert-\vert 3-x\vert=\vert x-4\vert$;
  8. $ \vert x-1\vert+\vert 1-2x\vert=2\vert x\vert$;
  9. $ \vert x\vert-2\vert x+1\vert+3\vert x+2\vert=0$;
  10. $ \vert x+1\vert-\vert x\vert+3\vert x-1\vert=x+2$;
  11. $ \vert x\vert-2\vert x+1\vert+3\vert x+2\vert=0$;
  12. $ \vert x\vert+2\vert x+1\vert-3\vert x-3\vert=0$;
  13. $ \vert x^2-9\vert+\vert x-2\vert=5$;
  14. $ \vert x^2-1\vert+x+1=0$;
  15. $ \vert x^2-4\vert-\vert 9-x^2\vert=5$;
  16. $ \vert x^2-9\vert+\vert x^2-4\vert=5$;
  17. $ x^2+4\vert x-3\vert-7x+11=0$;
  18. $ x^2-4\vert x+1\vert+5x+3=0$;
  19. $ \vert 2x-1\vert=\vert x+1\vert+\vert x-2\vert$;
  20. $ \vert x+2\vert-\vert x+3\vert+\vert x-4\vert=6$;
  21. $ \vert x\vert+3\vert x+1\vert=\vert x-6\vert$;
  22. $ \vert x\vert+2\vert x+1\vert=\vert x-5\vert$;
  23. $ \vert x-4\vert+\vert x+1\vert-\vert x+3\vert=6$;
  24. $ \vert x-7\vert-\vert x+3\vert+\vert x-5\vert=-1$;
  25. $ \vert 2x-1\vert+\vert x+2\vert-\vert 4x\vert=2$.


I.3.Atrisināt nevienādības.

  1. $ x^2-3\vert x\vert+2>0$;
  2. $ x^2-\vert 5x-3\vert-x<2$;
  3. $ x^2+5\vert x\vert-24>0$;
  4. $ \vert x^2-3x-15\vert<2x^2-x$;
  5. $ \vert x^2+x+10\vert\leq 3x^2+7x+2$;
  6. $ \vert 2x^2+x+11\vert>x^2-5x+6$;
  7. $ 3x^2-\vert x-3\vert<x-2$;
  8. $ \vert x-6\vert>\vert x^2-5x+9\vert$;
  9. $ \vert x\vert+\vert x-1\vert>5$;
  10. $ \vert x+1\vert+\vert x-2\vert>5$;
  11. $ \vert 2x+1\vert+\vert 5x-2\vert\geq 1$;
  12. $ \vert 3x-1\vert+\vert 2x-3\vert-x<2$;
  13. $ \vert x-1\vert+\vert 2-x\vert>3+x$;
  14. $ \vert 4-x\vert-\vert 2-3x\vert\geq 4$;
  15. $ \vert x-3\vert+\vert 2x+1\vert<8$;
  16. $ \vert 2x-5\vert-\vert 3x+6\vert>-3$;
  17. $ \vert 5-3x\vert-\vert 3x-1\vert\leq 2$;
  18. $ \vert 1-3x\vert-\vert 2x+3\vert\geq 0$;
  19. $ \vert x+3\vert-\vert x+1\vert<2$;
  20. $ \vert x+2\vert+\vert x-2\vert\leq 12$;
  21. $ \vert x-2\vert\leq
2x^2-9x+9$;
  22. $ \vert x+3\vert+\vert x-2\vert<13$;
  23. $ 2\vert x+3\vert-3\vert x-2\vert>6$;
  24. $ \vert x+3\vert-\vert x-2\vert>2x-3$;
  25. $ \vert x-2\vert+3\vert x+3\vert>4x+25$.


II.Noteikt ar formulu uzdotās funkcijas definīcijas apgabalu.

  1. $ f(x)=\sqrt{x+7}+\frac{1}{\lg(3+x)}$;
  2. $ f(x)=\arccos(2+x)-\sqrt{\frac{2+x}{1-x}}$;
  3. $ f(x)=\lg(x^2-5x+4)+\frac{1}{\sqrt{25-x^2}}$;
  4. $ f(x)=\arccos\frac{1-3x}{4}+\sqrt{1-x^2}$;
  5. $ f(x)=\frac{1}{\sqrt[5]{3x-4}}+\sqrt{\lg\frac{3x}{5-2x}}$;
  6. $ f(x)=\sqrt{\lg\frac{3x+1}{2x-1}}-\arcsin(1-x)$;
  7. $ f(x)=\frac{x+7}{\lg(3x+2)}+\arccos\frac{5x}{7}$;
  8. $ f(x)=\lg\frac{3x-8}{5}+3^{\frac{x}{7-x}}$;
  9. $ f(x)=\sqrt[3]{\frac{1}{3x-2}}+\frac{\sqrt{2x+5}}{\log_2(3-x)}$;
  10. $ f(x)=\arcsin\frac{3-2x}{5}+\sqrt{3-x}$;
  11. $ f(x)=\sqrt{\log_{0,5}(9-2x)}-2^{\frac{\cos x}{2x-4}}$;
  12. $ f(x)=\sqrt{x+7}+\frac{1}{\lg(3+x)}$;
  13. $ f(x)=\frac{\sqrt{16-x^2}}{\log_2(x+1)-1}+3^{\frac{x+1}{x}}$;
  14. $ f(x)=\frac{\sqrt{5x+1}}{\log_{0,3}(4-x)}-\sqrt{\frac{1}{x-2}}$;
  15. $ f(x)=\sqrt{\lg\frac{5x-8}{3}}+4^{\frac{3x}{6-x}}$;
  16. $ f(x)=\lg(36-x^2)+2\arccos\frac{5}{x}$;
  17. $ f(x)=2^{\frac{\cos
x}{12-3x}}+3\sqrt{\log_{0,2}(5-3x)}$;
  18. $ f(x)=\frac{5x+7}{\lg(7-x)}-\frac{7}{(x+3)^2}$;
  19. $ f(x)=\arcsin\frac{x}{5}-\frac{3x-4}{\log_5(4x-7)}$;
  20. $ f(x)=\sqrt[4]{\log_{0,3}(1-4x)}+2\arctg\frac{\sqrt[3]{8-3x}}{24x^2-1}$;
  21. $ f(x)=\sqrt{\lg\frac{5x-18}{3}}-2^{\frac{x}{6x-x^2}}$;
  22. $ f(x)=\frac{2x+3}{\sqrt{3x-x^2}}-\arcsin\frac{2}{3x-1}$;
  23. $ f(x)=\frac{15x^2-1}{\sqrt{5x^2+7x}}-\lg(x^2-6x)$;
  24. $ f(x)=\sqrt{3-x}+\arcsin\frac{3-2x}{5}$;
  25. $ f(x)=\arcsin\frac{x-3}{2}-\lg(4-x)$.


III.Noteikt, kuras no dotajām funkcijām ir pāra funkcijas, kuras - nepāra funkcijas.

  1. $ f(x)=\frac{\vert\sin x\vert-3}{\tg x^3}$;
  2. $ f(x)=\frac{1-4^x}{4^x+1}$;
  3. $ f(x)=\lg\left(2x+\sqrt{4x^2+1}\right)$;
  4. $ f(x)=\lg\frac{2+x}{2-x}+x\tg^2x$;
  5. $ f(x)=\lg\frac{1+x}{1-x}-4$;
  6. $ f(x)=\frac{1}{3^x}-\frac{1}{3^{-x}}$;
  7. $ f(x)=\frac{1}{2}(e^x+e^{-x})$;
  8. $ f(x)=\sqrt{1+x+x^2}-\sqrt{1-x+x^2}$;
  9. $ f(x)=\frac{e^{3x}-e^{-3x}}{2}$;
  10. $ f(x)=\frac{1}{2}\lg
x^2+2-x\sin 3x$;
  11. $ f(x)=\frac{e^{2x}+1}{e^{-2x}-1}$;
  12. $ f(x)=x\frac{e^x-1}{e^x+1}$;
  13. $ f(x)=\tg^2x-\cos x+4$;
  14. $ f(x)=x\sqrt[3]{x}-\ctg 2x$;
  15. $ f(x)=\sqrt[3]{(x+1)^2}+\sqrt[3]{(x-1)^2}$;
  16. $ f(x)=x\sqrt{x^2}-\ctg 2x+3x$;
  17. $ f(x)=\tg^3x-2x^4-\sin 2x$;
  18. $ f(x)=x^2+\vert x\vert-2$;
  19. $ f(x)=6x^5-2\sin x$;
  20. $ f(x)=2\tg x+\sin 2x$;
  21. $ f(x)=\frac{2^{\vert x\vert}}{x^2}-3$;
  22. $ f(x)=\frac{(2^{2x}-1)x}{2^x}$;
  23. $ f(x)=\frac{(x-2)^2\ctg^5x+1}{1-\cos 3x}$;
  24. $ f(x)=2\sin\frac{x}{2}-x+x^3$;
  25. $ f(x)=\frac{\sin
x}{x}+x^5\ctg^2x$.


IV.Noteikt, vai dotās funkcijas ir periodiskas un noteikt periodu.

  1. $ f(x)=\sin\frac{\pi x}{2}+1$;
  2. $ f(x)=a\sin 4x+b\cos 3x\;\;(a,b\in\mathbb{R})$;
  3. $ f(x)=\sin\frac{x}{2}+\ctg x$;
  4. $ f(x)=\sin\frac{5x}{2}-\cos
x$;
  5. $ f(x)=\sin\frac{3x}{2}+1-\tg x$;
  6. $ f(x)=\sin
2x-2\tg\frac{x}{2}$;
  7. $ f(x)=\cos\frac{\pi x}{4}-\sin\frac{\pi
x}{6}$;
  8. $ f(x)=\sin\frac{\pi x}{3}+\sin\frac{\pi x}{4}$;
  9. $ f(x)=2\sin 3x+3\sin x$;
  10. $ f(x)=\sin x+\cos 2x$;
  11. $ f(x)=\sin\left(2\pi
x+\frac{\pi}{3}\right)+2\sin\left(3\pi+\frac{\pi}{4}\right)+3\sin\pi
x$;
  12. $ f(x)=1+\tg 3x-\cos 2x$;
  13. $ f(x)=10\sin 3x+\tg 5x$;
  14. $ f(x)=\sqrt{\tg 3x}-\sin 3x$;
  15. $ f(x)=3\sin\frac{5x}{8}+2\cos\frac{7x}{8}$;
  16. $ f(x)=\sin\frac{3\pi x}{4}-\cos\frac{3\pi x}{2}$;
  17. $ f(x)=2\sin\frac{x}{2}+3\tg\frac{x}{2}$;
  18. $ f(x)=\sin\frac{2x+3}{6\pi}$;
  19. $ f(x)=-\cos\frac{x-1}{2}$;
  20. $ f(x)=2\sin(3x+5)$;
  21. $ f(x)=4\sin\frac{\pi x+1}{3}$;
  22. $ f(x)=1+\tg\left(\frac{5x}{6}+1\right)$;
  23. $ f(x)=\sin\frac{x}{3}+\ctg\frac{x}{4}$;
  24. $ f(x)=\frac{1}{\cos
2x}+\tg\frac{x}{2}$;
  25. $ f(x)=\sin 2x+2\sin 3x$.


V. Katrai no dotajām funkcijām atrast apvērsto funkciju un noteikt apvērstās funkcijas definīcijas un vērtību apgabalu.

  1. $ f(x)=\arcsin\frac{3}{2+x}$;
  2. $ f(x)=\sqrt[3]{x+1}$;
  3. $ f(x)=\arctg 3x$;
  4. $ f(x)=\lg(x-1)$;
  5. $ f(x)=\cos^3x,\;\;x\in[0;\pi]$;
  6. $ f(x)=\sqrt[3]{1-x^3}$;
  7. $ f(x)=5^{\cos x},\;\;x\in[0;\pi]$;
  8. $ f(x)=\cos(3x-2),\;\;x\in\left[\frac{2}{3};\frac{\pi}{3}+\frac{2}{3}\right]$;
  9. $ f(x)=\log_2\frac{x-5}{4}$;
  10. $ f(x)=\arcsin\frac{3}{4x-1}$;
  11. $ f(x)=1+\lg(x+2)$;
  12. $ f(x)=\frac{2^x}{1+2^x}$;
  13. $ f(x)=2\sin
3x,\;\;x\in\left[-\frac{\pi}{6};\frac{\pi}{5}\right]$;
  14. $ f(x)=\frac{10^x-10^{-x}}{10^x+10^{-x}}$;
  15. $ f(x)=10^{x+1}$;
  16. $ f(x)=\log_2(1-x^2),\;\;x>0$;
  17. $ f(x)=\lg\frac{x}{2}$;
  18. $ f(x)=\log_5\frac{4}{x}$;
  19. $ f(x)=\sqrt[3]{x^3-27}$;
  20. $ f(x)=\frac{2x}{1+x}$;
  21. $ f(x)=7+\ln(x+14)$;
  22. $ f(x)=\sin^3x,\;\;x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$;
  23. $ f(x)=\ln(x^2-1),\;\;x>0$;
  24. $ f(x)=\sqrt[3]{x^3-125}$;
  25. $ f(x)=\lg(x^3-1)$.


VI.Lietojot virknes robežas definīciju, pierādīt, ka $ \lim\limits_{n\rightarrow\infty}a_n=a$. Skaitlim $ \varepsilon=0,01$ atrast atbilstošo $ N$.

$\displaystyle 1.$ $\displaystyle \;\;a_n=\frac{4+2n}{1-3n},\;\;a=-\frac{2}{3};$ $\displaystyle 2.$ $\displaystyle \;\;a_n=\frac{3n-2}{2n-1},\;\;a=\frac{3}{2};$    
$\displaystyle 3.$ $\displaystyle \;\;a_n=\frac{7n-1}{n+1},\;\;a=7;$ $\displaystyle 4.$ $\displaystyle \;\;a_n=\frac{n}{2n+1},\;\;a=\frac{1}{2};$    
$\displaystyle 5.$ $\displaystyle \;\;a_n=\frac{5^n-1}{5^n},\;\;a=1;$ $\displaystyle 6.$ $\displaystyle \;\;a_n=\frac{5n+3}{2n-1},\;\;a=\frac{5}{2};$    
$\displaystyle 7.$ $\displaystyle \;\;a_n=\frac{n^2+4}{5n^2-2},\;\;a=\frac{1}{5};$ $\displaystyle 8.$ $\displaystyle \;\;a_n=\frac{3n^2+1}{n^2+4},\;\;a=3;$    
$\displaystyle 9.$ $\displaystyle \;\;a_n=\frac{n-2}{n+2},\;\;a=1;$ $\displaystyle 10.$ $\displaystyle \;\;a_n=\frac{7n+4}{2n+1},\;\;a=\frac{7}{2};$    
$\displaystyle 11.$ $\displaystyle \;\;a_n=\frac{3n^2+1}{5-n^2},\;\;a=-3;$ $\displaystyle 12.$ $\displaystyle \;\;a_n=\frac{5n-2}{7n+3},\;\;a=\frac{5}{7};$    
$\displaystyle 13.$ $\displaystyle \;\;a_n=\frac{5n}{n+1},\;\;a=5;$ $\displaystyle 14.$ $\displaystyle \;\;a_n=\frac{4n+3}{8n-1},\;\;a=\frac{1}{2};$    
$\displaystyle 15.$ $\displaystyle \;\;a_n=\frac{n+2}{n+1};\;\;a=1;$ $\displaystyle 16.$ $\displaystyle \;\;a_n=\frac{3-n^2}{1+2n^2},\;\;a=-\frac{1}{2};$    
$\displaystyle 17.$ $\displaystyle \;\;a_n=\frac{5n^2-3}{3n^2+1},\;\;a=\frac{5}{3};$ $\displaystyle 18.$ $\displaystyle \;\;a_n=\frac{1-2n^2}{2-4n^2},\;\;a=\frac{1}{2};$    
$\displaystyle 19.$ $\displaystyle \;\;a_n=\frac{4^n}{2-4^n},\;\;a=-1;$ $\displaystyle 20.$ $\displaystyle \;\;a_n=\frac{5-3n}{7n+4},\;\;a=-\frac{3}{7};$    
$\displaystyle 21.$ $\displaystyle \;\;a_n=\frac{4-7n}{5-n},\;\;a=7;$ $\displaystyle 22.$ $\displaystyle \;\;a_n=\frac{5n-8}{3n+11},\;\;a=\frac{5}{3};$    
$\displaystyle 23.$ $\displaystyle \;\;a_n=\frac{2^n-1}{2^n},\;\;a=1;$ $\displaystyle 24.$ $\displaystyle \;\;a_n=\frac{2n-3}{4n+5},\;\;a=\frac{1}{2};$    
$\displaystyle 25.$ $\displaystyle \;\;a_n=\frac{5n+6}{n+1},\;\;a=5.$      


VII.Lietojot funkcijas robežas definīciju, pierādīt, ka $ \lim\limits_{x\rightarrow a}f(x)=A$. Skaitlim $ \varepsilon=0,01$ atrast atbilstošo $ \delta$.

$\displaystyle 1.$ $\displaystyle \;\;f(x)=3x^2-2,$   $\displaystyle \;\;a=-2,$   $\displaystyle \;\;A=10;$    
$\displaystyle 2.$ $\displaystyle \;\;f(x)=3-\frac{x}{5},$   $\displaystyle \;\;a=-5,$   $\displaystyle \;\;A=4;$    
$\displaystyle 3.$ $\displaystyle \;\;f(x)=x^2-x+2,$   $\displaystyle \;\;a=1,$   $\displaystyle \;\;A=2;$    
$\displaystyle 4.$ $\displaystyle \;\;f(x)=x^2+2x+3,$   $\displaystyle \;\;a=0,$   $\displaystyle \;\;A=3;$    
$\displaystyle 5.$ $\displaystyle \;\;f(x)=x^2+x+1,$   $\displaystyle \;\;a=3,$   $\displaystyle \;\;A=13;$    
$\displaystyle 6.$ $\displaystyle \;\;f(x)=3x-2,$   $\displaystyle \;\;a=-3,$   $\displaystyle \;\;A=-11;$    
$\displaystyle 7.$ $\displaystyle \;\;f(x)=3x+2,$   $\displaystyle \;\;a=2,$   $\displaystyle \;\;A=8;$    
$\displaystyle 8.$ $\displaystyle \;\;f(x)=5-2x,$   $\displaystyle \;\;a=0,$   $\displaystyle \;\;A=5;$    
$\displaystyle 9.$ $\displaystyle \;\;f(x)=x^2+x-5,$   $\displaystyle \;\;a=3,$   $\displaystyle \;\;A=7;$    
$\displaystyle 10.$ $\displaystyle \;\;f(x)=3x^2-1,$   $\displaystyle \;\;a=2,$   $\displaystyle \;\;A=11;$    
$\displaystyle 11.$ $\displaystyle \;\;f(x)=3-5x^2,$   $\displaystyle \;\;a=2,$   $\displaystyle \;\;A=-17;$    
$\displaystyle 12.$ $\displaystyle \;\;f(x)=x^2+4,$   $\displaystyle \;\;a=1,$   $\displaystyle \;\;A=5;$    
$\displaystyle 13.$ $\displaystyle \;\;f(x)=3x-2,$   $\displaystyle \;\;a=-1,$   $\displaystyle \;\;A=-5;$    
$\displaystyle 14.$ $\displaystyle \;\;f(x)=x^2-3x,$   $\displaystyle \;\;a=3,$   $\displaystyle \;\;A=0;$    
$\displaystyle 15.$ $\displaystyle \;\;f(x)=2x^2+1,$   $\displaystyle \;\;a=-1,$   $\displaystyle \;\;A=3;$    
$\displaystyle 16.$ $\displaystyle \;\;f(x)=x^2-7x+1,$   $\displaystyle \;\;a=7,$   $\displaystyle \;\;A=1;$    
$\displaystyle 17.$ $\displaystyle \;\;f(x)=x^2-6x+3,$   $\displaystyle \;\;a=6,$   $\displaystyle \;\;A=3;$    
$\displaystyle 18.$ $\displaystyle \;\;f(x)=3x-5,$   $\displaystyle \;\;a=2,$   $\displaystyle \;\;A=1;$    
$\displaystyle 19.$ $\displaystyle \;\;f(x)=x^2-8x+2,$   $\displaystyle \;\;a=-1,$   $\displaystyle \;\;A=11;$    
$\displaystyle 20.$ $\displaystyle \;\;f(x)=x^2+4x+2,$   $\displaystyle \;\;a=-2,$   $\displaystyle \;\;A=-12;$    
$\displaystyle 21.$ $\displaystyle \;\;f(x)=2x^2+1,$   $\displaystyle \;\;a=-2,$   $\displaystyle \;\;A=9;$    
$\displaystyle 22.$ $\displaystyle \;\;f(x)=4x+7,$   $\displaystyle \;\;a=-3,$   $\displaystyle \;\;A=-5;$    
$\displaystyle 23.$ $\displaystyle \;\;f(x)=8-5x,$   $\displaystyle \;\;a=0,$   $\displaystyle \;\;A=8;$    
$\displaystyle 24.$ $\displaystyle \;\;f(x)=x^2+2x-3,$   $\displaystyle \;\;a=-1,$   $\displaystyle \;\;A=-4;$    
$\displaystyle 25.$ $\displaystyle \;\;f(x)=x^2-2x-6,$   $\displaystyle \;\;a=-2,$   $\displaystyle \;\;A=2.$    


VIII.1.Atrast robežas.

$\displaystyle 1.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\frac{x^2-5x+6}{x^3-8x+8};$ $\displaystyle 2.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -3}\frac{x^2+2x-3}{3x^2+14x+15};$    
$\displaystyle 3.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\frac{x^2-2x+1}{x^3-x};$ $\displaystyle 4.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\frac{x^3+x-2}{x^3-x^2-x+1};$    
$\displaystyle 5.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{1}{2}}\frac{8x^3-1}{6x^2-5x+1};$ $\displaystyle 6.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -2}\frac{x^3+3x^2+2x}{x^2-x-6};$    
$\displaystyle 7.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 3}\frac{x^2-5x+6}{x^2-8x+15};$ $\displaystyle 8.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\frac{x^3-3x+2}{x^4-4x+3};$    
$\displaystyle 9.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\frac{x^4-3x+2}{x^5-4x+3};$ $\displaystyle 10.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\frac{x^3-2x^2-4x+8}{x^4-8x^2+16};$    
$\displaystyle 11.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -1}\frac{x^3-2x-1}{x^5-2x-1};$ $\displaystyle 12.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{3}{2}}\frac{2x^2-x-3}{2x^2-5x+3};$    
$\displaystyle 13.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{1}{3}}\frac{3x^2+5x-2}{1-2x-3x^2};$ $\displaystyle 14.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -1}\frac{x^2-4x-5}{x^2-2x-3};$    
$\displaystyle 15.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -2}\frac{2-3x-2x^2}{x^2+x-2};$ $\displaystyle 16.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 3}\frac{3x^2-11x+6}{2x^2-5x-3};$    
$\displaystyle 17.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\frac{x^3-3x-2}{x^3-8};$ $\displaystyle 18.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{4x^2-3x}{2x^2-9x};$    
$\displaystyle 19.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\frac{x^2-5x+6}{x^3-2x^2-x+2};$ $\displaystyle 20.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{x-3x^2}{5x^3-6x^2+x};$    
$\displaystyle 21.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{1}{3}}\frac{3x^2+2x-1}{27x^3-1};$ $\displaystyle 22.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 5}\frac{2x^2-11x+5}{3x^2-14x-5};$    
$\displaystyle 23.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -2}\frac{x^3+5x^2+8x+4}{x^3+3x^2-4};$ $\displaystyle 24.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -1}\frac{x^3+4x^2+5x+2}{x^3-3x-2};$    
$\displaystyle 25.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\frac{x^3-3x+2}{x^3-x^2-x+1}.$      


VIII.2.Atrast robežas.

$\displaystyle 1.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{x-3x^2}{4-\sqrt{x+16}};$ $\displaystyle 2.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\frac{3x-2-\sqrt{4x^2-x-2}}{x^2-3x+2};$    
$\displaystyle 3.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\frac{\sqrt[3]{x-2}-\sqrt[3]{1-x+x^2}}{x^2-1};$ $\displaystyle 4.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\frac{x-\sqrt{x+2}}{\sqrt{4x+1}-3};$    
$\displaystyle 5.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sqrt{x+1}-\sqrt{x^2+x+1}}{x^2};$ $\displaystyle 6.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sqrt[3]{1+x^2}-1}{x^2};$    
$\displaystyle 7.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sqrt{x^2+1}-1}{\sqrt{x^2+16}-4};$ $\displaystyle 8.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 5}\frac{\sqrt{x-1}-2}{x^2-25};$    
$\displaystyle 9.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sqrt[3]{1+x}-\sqrt[3]{1-x}}{x};$ $\displaystyle 10.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 4}\frac{\sqrt{1+2x}-3}{\sqrt{x}-2};$    
$\displaystyle 11.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -8}\frac{\sqrt{1-x}-3}{2+\sqrt[3]{x}};$ $\displaystyle 12.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 3}\frac{\sqrt{x+13}-2\sqrt{x+1}}{x^2-9};$    
$\displaystyle 13.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 16}\frac{\sqrt[4]{x}-2}{\sqrt{x}-4};$ $\displaystyle 14.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -2}\frac{\sqrt[3]{x-6}+2}{x^3+8};$    
$\displaystyle 15.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 8}\frac{\sqrt{9+2x}-5}{\sqrt[3]{x}-2};$ $\displaystyle 16.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sqrt{1-2x-x^2}-1-x}{x};$    
$\displaystyle 17.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sqrt[3]{8+3x-x^2}-2}{x+x^2};$ $\displaystyle 18.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sqrt[3]{27+x}-\sqrt[3]{27-x}}{x+2\sqrt[3]{x^4}};$    
$\displaystyle 19.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 3}\frac{\sqrt{2x+3}-3}{9-x^2};$ $\displaystyle 20.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\frac{\sqrt{x}-\sqrt{2-x}}{1-x^2};$    
$\displaystyle 21.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{x}{\sqrt{x+4}-2};$ $\displaystyle 22.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -9}\frac{4-\sqrt{7-x}}{\sqrt{10+x}-1};$    
$\displaystyle 23.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\frac{\sqrt[3]{x}-1}{2-\sqrt{x+3}};$ $\displaystyle 24.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -1}\frac{\sqrt{3x+7}-2}{1+\sqrt[3]{x}};$    
$\displaystyle 25.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 4}\frac{x-\sqrt{3x+4}}{16-x^2}.$      


VIII.3.Atrast robežas.

$\displaystyle 1.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{3x^3-9x^2+13x+1}{4x^3+8x^2-7x+16};$ $\displaystyle 2.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{4x^3+8x+7}{9x^3+11x+3};$    
$\displaystyle 3.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{13x^2-2x+5}{x^4+6x+1};$ $\displaystyle 4.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{x^5-12x^3+7}{4x^4+8x^3+11};$    
$\displaystyle 5.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{x^2-2x+5}{x^3+3x+7};$ $\displaystyle 6.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{2x^2+7x-1}{3x^2-5x+6};$    
$\displaystyle 7.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{6x^2-7x-12}{2x^2-5x-8};$ $\displaystyle 8.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{x^2-4x+3}{x+5};$    
$\displaystyle 9.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{x^4-2x^3-1}{100x^3+2x^2};$ $\displaystyle 10.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{3x^3-4x^2+8}{-5x^3+2x^2+x};$    
$\displaystyle 11.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{5+3x^3}{7x^4-2x^2+1};$ $\displaystyle 12.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{2-x-4x^2}{x^3-7x+8};$    
$\displaystyle 13.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{3x^2-2x-1}{7x-9};$ $\displaystyle 14.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{x^5-3x^4+7x-1}{3x^5+2x^3-3};$    
$\displaystyle 15.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{3x^2-2x+10}{5x^3+3x-1};$ $\displaystyle 16.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{2x^3-3x^2+5}{3x^4-7x+1};$    
$\displaystyle 17.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{3x^2-1}{5x^2+2x};$ $\displaystyle 18.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{6x^2+5x-1}{3x^2-x+1};$    
$\displaystyle 19.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{1-x-x^2}{x^3+3};$ $\displaystyle 20.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{x^4-2x^2+3}{3x^3-5};$    
$\displaystyle 21.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{x^2-1}{2x^2-x-1};$ $\displaystyle 22.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{2x^2+x-1}{x^2-5x+6};$    
$\displaystyle 23.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{x^2-9x+14}{x^3+3x+14};$ $\displaystyle 24.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{10x^2+3x+1}{x^3-x^2-x};$    
$\displaystyle 25.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\frac{x^2-5x-36}{2x^2-10x+12}.$      


VIII.4.Atrast robežas.

  1. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+7x+9}-\sqrt{x^2+3x+5}\right)$;
  2. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+x+1}-\sqrt{x^2-x+1}\right)$;
  3. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^4-3x+1}-x^2\right)$;
  4. $ \lim\limits_{x\rightarrow+\infty}\left(2x-1-\sqrt{4x^2-4x-3}\right)$;
  5. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2-2x-1}-\sqrt{x^2-7x+3}\right)$;
  6. $ \lim\limits_{x\rightarrow\infty}\left(\sqrt{x^2+1}-\sqrt{x^2+3}\right)$;
  7. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+1}-x\right)$;
  8. $ \lim\limits_{x\rightarrow\infty}\left(\sqrt[3]{x^3+5x}-\sqrt[3]{x^3+8x}\right)$;
  9. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x+5}-\sqrt{x}\right)$;
  10. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{2x+1}-\sqrt{x+2}\right)$;
  11. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{4x-1}-x\right)$;
  12. $ \lim\limits_{x\rightarrow+\infty}\left(x-\sqrt{x^2+7x}\right)$;
  13. $ \lim\limits_{x\rightarrow-\infty}\left(\sqrt{3-2x}+x\right)$;
  14. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+10x}-x\right)$;
  15. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{4x^2+3x}-2x\right)$;
  16. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2-2x+1}-\sqrt{x^2+7x+3}\right)$;
  17. $ \lim\limits_{x\rightarrow\infty}\left(\sqrt[3]{(x+1)^2}-\sqrt[3]{(x-1)^2}\right)$;
  18. $ \lim\limits_{x\rightarrow+\infty}x^{\frac{3}{2}}\left(\sqrt{x^3+1}-\sqrt{x^3-1}\right)$;
  19. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2-1}-\sqrt{x^2-x-1}\right)$;
  20. $ \lim\limits_{x\rightarrow-\infty}x\left(\sqrt{x^2+1}+x\right)$;
  21. $ \lim\limits_{x\rightarrow\infty}\left(\sqrt[3]{x^3+x^2+1}-\sqrt[3]{x^3-x^2+1}\right)$;
  22. $ \lim\limits_{x\rightarrow-\infty}\left(x+\sqrt{x^2-2x}\right)$;
  23. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt[3]{x^3+2x^2}-x\right)$;
  24. $ \lim\limits_{x\rightarrow\infty}\left(\sqrt{2x^2+1}+\sqrt{x^2-1}\right)$;
  25. $ \lim\limits_{x\rightarrow+\infty}\left(\sqrt{x^2+2x}-\sqrt{x^2+x}\right)$.


VIII.5.Atrast robežas.

$\displaystyle 1.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\cos^3x}{x\sin 2x};$ $\displaystyle 2.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1+\sin x-\cos x}{1-\sin x-\cos x};$    
$\displaystyle 3.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\left(\frac{1}{\sin x}-\frac{1}{\tg x}\right);$ $\displaystyle 4.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{2}}\frac{\cos x}{\sqrt[3]{(1-\sin x)^2}};$    
$\displaystyle 5.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{6}}\frac{\sin\left(x-\frac{\pi}{6}\right)}{\frac{\sqrt{3}}{2}-\cos x};$ $\displaystyle 6.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{4}}\frac{\cos x-\sin x}{\cos 2x};$    
$\displaystyle 7.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{4}}\frac{\sin x-\cos x}{1-\tg x};$ $\displaystyle 8.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\sqrt{\cos x}}{x^2};$    
$\displaystyle 9.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{x-\sin 2x}{x+\sin 3x};$ $\displaystyle 10.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\tg-\sin x}{\sin 3x};$    
$\displaystyle 11.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sin 5x-\sin 3x}{\sin x};$ $\displaystyle 12.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\cos x-\cos3x}{x^2};$    
$\displaystyle 13.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{x\sin 3x}{1-\cos 3x};$ $\displaystyle 14.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{3}}\frac{\sin\left(x-\frac{\pi}{3}\right)}{1-2\cos x};$    
$\displaystyle 15.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\cos 4x}{\sin^27x};$ $\displaystyle 16.$ $\displaystyle \;\;\lim\limits_{x\rightarrow-\frac{\pi}{4}}\frac{1+\sin2x}{\sin x+\cos x};$    
$\displaystyle 17.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sin x-\tg x}{4\sin^2\frac{x}{2}};$ $\displaystyle 18.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{2}}\frac{\cos x}{\cos\frac{x}{2}-\sin\frac{x}{2}};$    
$\displaystyle 19.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{4}}\frac{1-\tg x}{\sin x-\cos x};$ $\displaystyle 20.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\cos 3x}{\tg^26x};$    
$\displaystyle 21.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\tg x-\sin x}{x^3};$ $\displaystyle 22.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\tg x}{\sin 5x-\sin x};$    
$\displaystyle 23.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\cos 5x}{x\tg 2x};$ $\displaystyle 24.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\cos 2x-1}{x\sin x};$    
$\displaystyle 25.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{3x^2-5x}{\sin^23x}.$      


VIII.6.Atrast robežas.

$\displaystyle 1.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{2}}(\sin x)^{\tg x};$ $\displaystyle 2.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{4}}(\tg x)^{\tg 2x};$    
$\displaystyle 3.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{x+1}{x-2}\right)^{2x-1};$ $\displaystyle 4.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{x^2+1}{x^2-2}\right)^{x^2};$    
$\displaystyle 5.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{4}}(\ctg x)^{\ctg 8x};$ $\displaystyle 6.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{2}}(\cos 4x)^{\tg x};$    
$\displaystyle 7.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{\pi}{2}}(\sin x)^{\ctg 4x};$ $\displaystyle 8.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{3x-4}{3x+2}\right)^{\frac{x+1}{3}};$    
$\displaystyle 9.$ $\displaystyle \;\;\lim\limits_{x\rightarrow2\pi}(\cos x)^{\ctg x};$ $\displaystyle 10.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{x+3}{x-4}\right)^{\frac{x}{2}-3};$    
$\displaystyle 11.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{x-7}{x-4}\right)^{\frac{x}{4}+5};$ $\displaystyle 12.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{x+5}{x+2}\right)^{\frac{x-1}{3}};$    
$\displaystyle 13.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{2x-1}{2x+1}\right)^{x+3};$ $\displaystyle 14.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{x-5}{x-2}\right)^{\frac{x}{3}-4};$    
$\displaystyle 15.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{5x-6}{5x-3}\right)^{\frac{x+1}{5}};$ $\displaystyle 16.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{3x-1}{3x+1}\right)^{x+4};$    
$\displaystyle 17.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{x^2-1}{x^2}\right)^x;$ $\displaystyle 18.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\left(1-x^2\right)^{\frac{1}{\tg 2x}};$    
$\displaystyle 19.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(1-\frac{1}{x^3}\right)^{x^2};$ $\displaystyle 20.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\left(1-\sin^2x\right)^{\frac{1}{x^2}};$    
$\displaystyle 21.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\left(1+x^3\right)^{\frac{1}{\sin x}};$ $\displaystyle 22.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\left(1-2x^2\right)^{\frac{1}{\sin^2x}};$    
$\displaystyle 23.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}(1+2x)^{\frac{1}{2\sin x}};$ $\displaystyle 24.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\left(1+x+x^3\right)^{\frac{1}{\sin x}};$    
$\displaystyle 25.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(\frac{x^2-2}{x^2+3}\right)^{x^2}.$      


VIII.7.Atrast robežas.

$\displaystyle 1.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{2^x-1}{\ln(1+2x)};$ $\displaystyle 2.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\ln(1+\sin x)}{\sin 4x};$    
$\displaystyle 3.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\cos 10x}{e^{x^2}-1};$ $\displaystyle 4.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\cos x}{\left(e^{3x}-1\right)^2};$    
$\displaystyle 5.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\ln(1+10x)}{\tg x};$ $\displaystyle 6.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\ln(1-5x)}{\sin x};$    
$\displaystyle 7.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{2^x-1}{\ln(x+1)};$ $\displaystyle 8.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\ln(1+\sin 2x)}{\sin 9x};$    
$\displaystyle 9.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\ln(1+\sin 5x)}{1-\cos 10x};$ $\displaystyle 10.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\cos 8x}{3^x-1};$    
$\displaystyle 11.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{5^{3x}-1}{e^x-1};$ $\displaystyle 12.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{(e^{-8x}-1)x}{1-\cos 4x};$    
$\displaystyle 13.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{3^x-1}{2^x-1};$ $\displaystyle 14.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\ln(1+3x)}{\sin 5x};$    
$\displaystyle 15.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{4^x-1}{\sin x};$ $\displaystyle 16.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{e^{3x}-1}{\sin x};$    
$\displaystyle 17.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\sin 2x}{\ln(1+x)};$ $\displaystyle 18.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{\ln(1+2x)}{\arcsin x};$    
$\displaystyle 19.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\cos 3x}{2\ln(1+9x^2)};$ $\displaystyle 20.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{1-\cos 6x}{\ln(1+3x^2)};$    
$\displaystyle 21.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{5^{x^2}-1}{4^{x^2}-1};$ $\displaystyle 22.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{5\sin 3x}{e^{-3x}-1};$    
$\displaystyle 23.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{x\sin 7x}{e^{-3x^2}-1};$ $\displaystyle 24.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{x\ln(1+4x)}{2\ln(1+2x^2)};$    
$\displaystyle 25.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\frac{xe^{\frac{x}{2}}-x}{\ln(1+2x^2)}.$      


VIII.8.Atrast robežas.

$\displaystyle 1.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 0}\left(\frac{x^3-3x+1}{x-4}+1\right);$ $\displaystyle 2.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^{\frac{x+1}{x}};$    
$\displaystyle 3.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\frac{\log_3(7-x^2)}{2-x-x^2};$ $\displaystyle 4.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 5}\left(x-\sqrt{x^2-5x}\right);$    
$\displaystyle 5.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 5}\left(1+\frac{4}{x}\right)^{\frac{x}{5}};$ $\displaystyle 6.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 3}\frac{\log_{19}(1+2x^2)}{4^x-62};$    
$\displaystyle 7.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -1}\frac{\log_5(2-3x)}{x^4};$ $\displaystyle 8.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\frac{x^2-x+1}{x-2};$    
$\displaystyle 9.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 3}\frac{x^2-3}{x^2+4x-9};$ $\displaystyle 10.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\left(\frac{3}{x^3+1}-\frac{1}{x+1}\right);$    
$\displaystyle 11.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 22}\frac{x-6}{\sqrt{x+3}-3};$ $\displaystyle 12.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\left(5+\frac{2}{x}-\frac{3}{x^2}\right);$    
$\displaystyle 13.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 4}\left(x-\sqrt{x^2-4x}\right);$ $\displaystyle 14.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 6}\left(1+\frac{3}{x}\right)^{\frac{x}{2}};$    
$\displaystyle 15.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -2}\frac{1+\sqrt{2x^2+1}}{x};$ $\displaystyle 16.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -3}\frac{6x^2+5x-1}{3x^2-x+8};$    
$\displaystyle 17.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 3}\frac{9-x^2}{x^3-5x+2};$ $\displaystyle 18.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\frac{x^3-4x+5}{x^2+6};$    
$\displaystyle 19.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 1}\left(\sqrt{5x-4}+\log_2x\right);$ $\displaystyle 20.$ $\displaystyle \;\;\lim\limits_{x\rightarrow\frac{1}{5}}\left(x^3-\log_{\frac{1}{5}}x\right);$    
$\displaystyle 21.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 4}\left(\frac{x-3}{x+1}\right)^{x-1};$ $\displaystyle 22.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 4}\left(x^2\sqrt{x+5}-x\sqrt{x}\right);$    
$\displaystyle 23.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 6}\frac{\sqrt[3]{4x+3}-x\sqrt{x-2}}{2x-3};$ $\displaystyle 24.$ $\displaystyle \;\;\lim\limits_{x\rightarrow 2}\frac{\log_5(1+2x)}{x^2};$    
$\displaystyle 25.$ $\displaystyle \;\;\lim\limits_{x\rightarrow -2}\frac{\log_{13}(1+3x^2)}{5^{-x}-1}.$      


IX.Salīdzināt dotās bezgalīgi mazās funkcijas.

$\displaystyle 1.$ $\displaystyle \;\;\alpha(x)=1-\sin x,$   $\displaystyle \;\;\beta(x)=\cos x,$   kad$\displaystyle \;\;x\rightarrow\frac{\pi}{2};$    
$\displaystyle 2.$ $\displaystyle \;\;\alpha(x)=\frac{x}{x-1},$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 3.$ $\displaystyle \;\;\alpha(x)=\tg x+x^2,$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 4.$ $\displaystyle \;\;\alpha(x)=2\sqrt{\sin x},$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 5.$ $\displaystyle \;\;\alpha(x)=1-\cos x,$   $\displaystyle \;\;\beta(x)=\sin x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 6.$ $\displaystyle \;\;\alpha(x)=x\sin^2x,$   $\displaystyle \;\;\beta(x)=x\sin x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 7.$ $\displaystyle \;\;\alpha(x)=\sin 2x-2\sin x,$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 8.$ $\displaystyle \;\;\alpha(x)=\frac{2x^4}{1+x},$   $\displaystyle \;\;\beta(x)=x^2,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 9.$ $\displaystyle \;\;\alpha(x)=5x^3+2x^2,$   $\displaystyle \;\;\beta(x)=3x^2+2x^3,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 10.$ $\displaystyle \;\;\alpha(x)=1-\left(\cos\sqrt{x}\right)^3,$   $\displaystyle \;\;\beta(x)=1-\cos x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 11.$ $\displaystyle \;\;\alpha(x)=\sqrt[3]{x},$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 12.$ $\displaystyle \;\;\alpha(x)=1-x,$   $\displaystyle \;\;\beta(x)=\sqrt{1-x},$   kad$\displaystyle \;\;x\rightarrow 1;$    
$\displaystyle 13.$ $\displaystyle \;\;\alpha(x)=\sin 2x+x^2,$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 14.$ $\displaystyle \;\;\alpha(x)=2-2\cos x,$   $\displaystyle \;\;\beta(x)=x^2,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 15.$ $\displaystyle \;\;\alpha(x)=\cos 2x,$   $\displaystyle \;\;\beta(x)=\sin 4x,$   kad$\displaystyle \;\;x\rightarrow \frac{\pi}{4};$    
$\displaystyle 16.$ $\displaystyle \;\;\alpha(x)=\frac{1-x}{1+x},$   $\displaystyle \;\;\beta(x)=1-\sqrt{x},$   kad$\displaystyle \;\;x\rightarrow 1;$    
$\displaystyle 17.$ $\displaystyle \;\;\alpha(x)=1-x,$   $\displaystyle \;\;\beta(x)=1-\sqrt[3]{x},$   kad$\displaystyle \;\;x\rightarrow 1;$    
$\displaystyle 18.$ $\displaystyle \;\;\alpha(x)=\sqrt[3]{x^2}-\sqrt{x},$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 19.$ $\displaystyle \;\;\alpha(x)=\sin 2x-\sin x,$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 20.$ $\displaystyle \;\;\alpha(x)=\frac{x(x+1)}{1-\sqrt{x}},$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 21.$ $\displaystyle \;\;\alpha(x)=\cos x-\sqrt[3]{\cos x},$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 22.$ $\displaystyle \;\;\alpha(x)=\frac{2x}{1+x},$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 23.$ $\displaystyle \;\;\alpha(x)=\sqrt{x+\sqrt{x}},$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 24.$ $\displaystyle \;\;\alpha(x)=e^{\sqrt{x}}-1,$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0;$    
$\displaystyle 25.$ $\displaystyle \;\;\alpha(x)=\sqrt{1+x}-1,$   $\displaystyle \;\;\beta(x)=x,$   kad$\displaystyle \;\;x\rightarrow 0.$    


X.Pierādīt, ka dotās funkcijas ir nepārtrauktas punktā $ x_0$.

$\displaystyle 1.$ $\displaystyle \;\;f(x)=4x^2+6,$   $\displaystyle \;\;x_0=7;$    
$\displaystyle 2.$ $\displaystyle \;\;f(x)=5x^2+x+9,$   $\displaystyle \;\;x_0=1;$    
$\displaystyle 3.$ $\displaystyle \;\;f(x)=42x+3x^2,$   $\displaystyle \;\;x_0=2;$    
$\displaystyle 4.$ $\displaystyle \;\;f(x)=x^3-x^2-4x+4,$   $\displaystyle \;\;x_0=3;$    
$\displaystyle 5.$ $\displaystyle \;\;f(x)=2x^2+2x-3,$   $\displaystyle \;\;x_0=-1;$    
$\displaystyle 6.$ $\displaystyle \;\;f(x)=2x^3+3x^2+5,$   $\displaystyle \;\;x_0=3;$    
$\displaystyle 7.$ $\displaystyle \;\;f(x)=x^3+4,$   $\displaystyle \;\;x_0=1;$    
$\displaystyle 8.$ $\displaystyle \;\;f(x)=x^2-5x+4,$   $\displaystyle \;\;x_0=2;$    
$\displaystyle 9.$ $\displaystyle \;\;f(x)=x^2-3x-4,$   $\displaystyle \;\;x_0=2;$    
$\displaystyle 10.$ $\displaystyle \;\;f(x)=x^2-x-6,$   $\displaystyle \;\;x_0=4;$    
$\displaystyle 11.$ $\displaystyle \;\;f(x)=x^3-3x^2-x-2,$   $\displaystyle \;\;x_0=-3;$    
$\displaystyle 12.$ $\displaystyle \;\;f(x)=x^3+x^2+4x-4,$   $\displaystyle \;\;x_0=1;$    
$\displaystyle 13.$ $\displaystyle \;\;f(x)=x^3-9x^2,$   $\displaystyle \;\;x_0=-2;$    
$\displaystyle 14.$ $\displaystyle \;\;f(x)=x^3+4,$   $\displaystyle \;\;x_0=1;$    
$\displaystyle 15.$ $\displaystyle \;\;f(x)=3x^2+4x-4,$   $\displaystyle \;\;x_0=\frac{1}{2};$    
$\displaystyle 16.$ $\displaystyle \;\;f(x)=2x^3+3x^2-7x-1,$   $\displaystyle \;\;x_0=0;$    
$\displaystyle 17.$ $\displaystyle \;\;f(x)=x^3-5x^2+1,$   $\displaystyle \;\;x_0=2;$    
$\displaystyle 18.$ $\displaystyle \;\;f(x)=2x^2-17x+35,$   $\displaystyle \;\;x_0=1;$    
$\displaystyle 19.$ $\displaystyle \;\;f(x)=2x^2-7x-15,$   $\displaystyle \;\;x_0=\frac{1}{7};$    
$\displaystyle 20.$ $\displaystyle \;\;f(x)=5x^3+2x^2+9,$   $\displaystyle \;\;x_0=-4;$    
$\displaystyle 21.$ $\displaystyle \;\;f(x)=7x^2+3,$   $\displaystyle \;\;x_0=\frac{1}{3};$    
$\displaystyle 22.$ $\displaystyle \;\;f(x)=2x^2-\frac{1}{3}x-\frac{2}{3},$   $\displaystyle \;\;x_0=3;$    
$\displaystyle 23.$ $\displaystyle \;\;f(x)=x^2-x-4,$   $\displaystyle \;\;x_0=-5;$    
$\displaystyle 24.$ $\displaystyle \;\;f(x)=x^2-8x+15,$   $\displaystyle \;\;x_0=4;$    
$\displaystyle 25.$ $\displaystyle \;\;f(x)=9x^3+1,$   $\displaystyle \;\;x_0=-3.$    


XI.Atrast funkcijas pārtraukuma punktus un noteikt to veidu. Konstruēt funkcijas shematisku grafiku.

$\displaystyle 1.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x^2+1, & \text{ja} & x\leq 1, \\ x-1, & \text{ja} & x>1; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{2}{x}, & \text{ja} & x<0...
... x<\frac{\pi}{2}, \\ 0, & \text{ja} & x\geq\frac{\pi}{2}; \\ \end{array}\right.$    
$\displaystyle 2.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x^2+1, & \text{ja} & x<1, \\ \frac{1}{x-1}, & \text{ja} & x>1; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \lg(1-x), & \text{ja} & x<1, \...
...\text{ja} & 1\leq x<2, \\ 3^{x-2}, & \text{ja} & x\geq 2; \\ \end{array}\right.$    
$\displaystyle 3.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x^2+1, & \text{ja} & x<1, \\ 3, & \text{ja} & x=1, \\ x+1, & \text{ja} & x>1; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} -\frac{8}{x}, & \text{ja} & x<...
...text{ja} & 0\leq x\leq 2, \\ \ln(x-2), & \text{ja} & x>2; \\ \end{array}\right.$    
$\displaystyle 4.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{1}{x+1}, & \text{ja} & x<3, \\ 0, & \text{ja} & x=3, \\ \ln(x-3), & \text{ja} & x>3; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{x^3-1}{x-1}, & \text{ja} & x\neq 1, \\ 3, & \text{ja} & x=1; \\ \end{array}\right.$    
$\displaystyle 5.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x^2-2, & \text{ja} & x<1, \\ -...
...ja} & 1\leq x<4, \\ \frac{1}{x-6}, & \text{ja} & x\geq 4; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} (x+1)^2, & \text{ja} & x\leq 0, \\ \ln x, & \text{ja} & x>0; \\ \end{array}\right.$    
$\displaystyle 6.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} 3^{x-2}, & \text{ja} & x<2, \\ x, & \text{ja} & 2\leq x<4, \\ \ln(x-4), & \text{ja} & x>4; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{25-x^2}{x-5}, & \text{ja} & x\neq 5, \\ -10, & \text{ja} & x=5; \\ \end{array}\right.$    
$\displaystyle 7.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} -8^{1-x}, & \text{ja} & x<1, \\ 0, & \text{ja} & x=1, \\ \frac{1}{x-3}, & \text{ja} & x>1; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \lg x, & \text{ja} & x>0, \\ 1-x^2, & \text{ja} & x\leq 0; \\ \end{array}\right.$    
$\displaystyle 8.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \tg x, & \text{ja} & \vert x\v...
...rac{1}{x^2}, & \text{ja} & \vert x\vert\geq\frac{\pi}{2}; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \sqrt{-x}, & \text{ja} & x<0, \\ e^x, & \text{ja} & x\geq 0; \\ \end{array}\right.$    
$\displaystyle 9.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{8-x^3}{x-2}, & \text{ja} & x\neq 2, \\ 0, & \text{ja} & x=2; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} -\frac{2}{x}, & \text{ja} & x<...
...text{ja} & 1\leq x\leq 3, \\ \lg(x-3), & \text{ja} & x>3; \\ \end{array}\right.$    
$\displaystyle 10.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x, & \text{ja} & x\leq 0, \\ 1...
...ext{ja} & 0<x\leq 1, \\ \frac{1}{1-x}, & \text{ja} & x>1; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} -\frac{9-x^2}{x+3}, & \text{ja} & x\neq -3, \\ 0, & \text{ja} & x=-3; \\ \end{array}\right.$    
$\displaystyle 11.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x^2, & \text{ja} & x<-1, \\ 1, & \text{ja} & x=-1, \\ \frac{1}{x+1}, & \text{ja} & x>-1; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{1}{2}, & \text{ja} & x<-...
...ja} & \vert x\vert\leq 1, \\ \lg(x-1), & \text{ja} & x>1; \\ \end{array}\right.$    
$\displaystyle 12.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{x^2-1}{x-1}, & \text{ja} & x<1, \\ 5, & \text{ja} & x=1; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \tg x, & \text{ja} & \vert x\v...
...\leq -\frac{\pi}{2}, \\ 1, & \text{ja} & x>\frac{\pi}{2}; \\ \end{array}\right.$    
$\displaystyle 13.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x^2+1, & \text{ja} & x<-2, \\ ...
...& \text{ja} & -2\leq x<2, \\ 3x+2, & \text{ja} & x\geq 2; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \sqrt{2-x}, & \text{ja} & x\leq 2, \\ \ln(x^2-4), & \text{ja} & x>2; \\ \end{array}\right.$    
$\displaystyle 14.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x+2, & \text{ja} & x<0, \\ 0, & \text{ja} & x=0, \\ 3-x^2, & \text{ja} & x>0; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \ln(2-x), & \text{ja} & x<2, \...
...xt{ja} & 2\leq x<3, \\ \sqrt{x-3}, & \text{ja} & x\geq 3; \\ \end{array}\right.$    
$\displaystyle 15.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x^3+1, & \text{ja} & x\leq 0, \\ \ln x, & \text{ja} & x>0; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{16-x^2}{x+4}, & \text{ja} & x\neq -4, \\ 1, & \text{ja} & x=-4; \\ \end{array}\right.$    
$\displaystyle 16.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} e^x, & \text{ja} & x<0, \\ \sq...
...x}, & \text{ja} & 0\leq x<3, \\ x^2-5, & \text{ja} & x>3; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{1}{x+1}, & \text{ja} & x...
...ja} & \vert x\vert\leq 1, \\ \lg(x-1), & \text{ja} & x>1; \\ \end{array}\right.$    
$\displaystyle 17.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{3}{x}, & \text{ja} & x\leq 3, \\ \ln(x-3), & \text{ja} & x>3; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x^2+2x, & \text{ja} & x<0, \\ ...
... \text{ja} & 0<x<1, \\ \sqrt{x-1}, & \text{ja} & x\geq 1; \\ \end{array}\right.$    
$\displaystyle 18.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x^2+3, & \text{ja} & x<0, \\ 3, & \text{ja} & x=0, \\ 4-x, & \text{ja} & x>0; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{1}{2-x}, & \text{ja} & x...
...ext{ja} & 0<x\leq 2, \\ \sqrt{-x}, & \text{ja} & x\leq 0; \\ \end{array}\right.$    
$\displaystyle 19.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} -\frac{1}{2}-x, & \text{ja} & ...
...c{\pi}{2}<x\leq 0, \\ x^2+\frac{1}{2}, & \text{ja} & x>0; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{x^3-8}{2-x}, & \text{ja} & x\neq 2, \\ 0, & \text{ja} & x=2; \\ \end{array}\right.$    
$\displaystyle 20.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \sin x, & \text{ja} & \vert x\...
...\\ \frac{1}{2}, & \text{ja} & \vert x\vert>\frac{\pi}{2}; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{4}{x}, & \text{ja} & x\l...
...ext{ja} & -2\leq x\leq 1, \\ \ln(x-1), & \text{ja} & x>1; \\ \end{array}\right.$    
$\displaystyle 21.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{1}{1-x}, & \text{ja} & x...
... x^2-4, & \text{ja} & 1<x\leq 3, \\ 5, & \text{ja} & x>3; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{x^2-36}{x+6}, & \text{ja} & x\neq -6, \\ 10, & \text{ja} & -2\leq x=-6; \\ \end{array}\right.$    
$\displaystyle 22.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} 3, & \text{ja} & x<-2, \\ x^2-1, & \text{ja} & -2\leq x<2, \\ x+4, & \text{ja} & x\geq 2; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \sqrt{1-x}, & \text{ja} & x\le...
... & \text{ja} & 1<x\leq 3, \\ \lg(x-3), & \text{ja} & x>3; \\ \end{array}\right.$    
$\displaystyle 23.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} 5-x, & \text{ja} & x<-1, \\ x^2+3, & \text{ja} & -1\leq x<2, \\ 7, & \text{ja} & x\geq 2; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \frac{1}{5+x}, & \text{ja} & x<-1, \\ 2, & \text{ja} & x=1, \\ 1-x^2, & \text{ja} & x<1; \\ \end{array}\right.$    
$\displaystyle 24.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} x-1, & \text{ja} & x<-2, \\ 4-x^2, & \text{ja} & -2\leq x<1, \\ 3, & \text{ja} & x\geq 1; \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \ln(5-x), & \text{ja} & x<5, \...
...\text{ja} & 5\leq x<6, \\ 2^{x-6}, & \text{ja} & x\geq 6; \\ \end{array}\right.$    
$\displaystyle 25.\;a)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} -x-11, & \text{ja} & x<-2, \\ ...
...& \text{ja} & -2\leq x<2, \\ x-11, & \text{ja} & x\geq 2. \\ \end{array}\right.$    
$\displaystyle b)$ $\displaystyle \;\;f(x)=\left\{\begin{array}{ccc} \tg x, & \text{ja} & \vert x\v...
...leq -\frac{\pi}{2}, \\ -x, & \text{ja} & x>\frac{\pi}{2}; \\ \end{array}\right.$    


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2003-05-15