L'Hopital's Rule

Evaluate the following limit, if it exists.

  1. \(\displaystyle{\lim_{x\to -1} \frac{x^9+1}{x^3+1}}\)    solution      

  2. \(\displaystyle{ \lim_{x\to 0} \frac{\sin x}{x}}\)    solution

  3. \(\displaystyle{ \lim_{x\to 0} \frac{1-\cos x}{x^2+2x}}\)   solution

  4. \(\displaystyle{ \lim_{x\to 2} \frac{x-2}{x^2-3x+2}}\)   solution

  5. \(\displaystyle{ \lim_{x\to 0} \frac{5x}{\sin^{-1}(x)}}\)    solution

  6. \(\displaystyle{ \lim_{x\to 0}  \frac{5^x-1}{5x}}\)   solution

  7. \(\displaystyle{ \lim_{\theta \to \pi/2} \frac{1+\cos(2\theta)}{1-\sin (\theta)}}\)    solution

  8. \(\displaystyle{ \lim_{x\to 0} \frac{x \, 3^x}{3^x -1}}\)  solution

  9. \(\displaystyle{ \lim_{x\to \infty} \frac{x^2+5}{e^{2x}}}\)  solution

  10. \(\displaystyle{ \lim_{x\to 0} \frac{x^3}{\sin x -x}}\)  solution

  11. \(\displaystyle{ \lim_{x\to 0} \frac{x-\tan x}{x^3}}\)  solution

  12. \(\displaystyle{ \lim_{x\to \infty} \frac{(\ln x)^2}{x}}\)    solution

  13. \(\displaystyle{ \lim_{x\to 0^+} x \ln x } \)   solution

  14. \(\displaystyle{ \lim_{x\to \infty} \: \frac{\ln(5x)}{\sqrt{5x}}}\)     solution

  15. \(\displaystyle{ \lim_{x\to -\infty} \, x^2\,e^{3x} }\)    solution

  16. \(\displaystyle{ \lim_{x\to \infty} \: x^3 e^{-x^5}}\)      solution

  17. \(\displaystyle{ \lim_{x\to 1} \: \frac{1+\cos(5\pi x)}{\ln x -x + 1} }\)     solution

  18. \(\displaystyle{ \lim_{x\to \infty} \: x \sin (5/x)}\)    solution

  19. \(\displaystyle{ \lim_{x\to \infty} \: x \tan(3/x)}\)   solution

  20. \(\displaystyle{ \lim_{x\to \infty} \: x^{\frac{1}{x}}}\)   solution