Activity 3.2.4

Evaluate each of the following limits. If you use L'Hôpital's Rule, indicate where it was used, and be certain its hypotheses are met before you apply it.

1. $\displaystyle \lim_{x \to \infty} \frac{x}{\ln(x)}$

2. $\displaystyle \lim_{x \to \infty} \frac{e^{x} + x}{2e^{x} + x^2}$

3. $\displaystyle \lim_{x \to 0^+} \frac{\ln(x)}{\frac{1}{x}}$

4. $\displaystyle \lim_{x \to \frac{\pi}{2}^-} \frac{\tan(x)}{x-\frac{\pi}{2}}$

5. $\displaystyle \lim_{x \to \infty} xe^{-x}$