Activity 3.2.2

Evaluate each of the following limits. If you use L'Hôpital's Rule, indicate where it was used, and be certain the limit is indeterminate before you apply it.

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

2. $\displaystyle \lim_{x \to \pi} \frac{\cos(x)}{x}$

3. $\displaystyle \lim_{x \to 1} \frac{2 \ln(x)}{1-e^{x-1}}$

4. $\displaystyle \lim_{x \to 0} \frac{\sin(x) - x}{\cos(2x)-1}$