L'Hospital'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.   \(\displaystyle{ \lim_{x\to \infty} \frac{(\ln x)^2}{x}}\)    solution      12. \(\displaystyle{ \lim_{x\to 0^+} x \ln x } \)   solution

 

13.  \(\displaystyle{ \lim_{x\to \infty} \: \frac{\ln(5x)}{\sqrt{5x}}}\)     solution      14. \(\displaystyle{ \lim_{x\to -\infty} \, x^2\,e^{3x} }\)    solution

 

15.   \(\displaystyle{ \lim_{x\to \infty} \: x^3 e^{-x^5}}\)      solution     16.  \(\displaystyle{ \lim_{x\to 1} \: \frac{1+\cos(5\pi x)}{\ln x -x + 1} }\)     solution

 

17.   \(\displaystyle{ \lim_{x\to \infty} \: x \sin (5/x)}\)    solution      18.   \(\displaystyle{ \lim_{x\to \infty} \: x \tan(3/x)}\)   solution


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