§3.2 Using derivatives to evaluate limits

Introduction

Using derivatives to evaluate indeterminate limits of the form

Limits involving

Determine the limit of the function

as $x \to \infty$.

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

Which of the following limits may legitimately be evaluated with L'Hopital's rule?

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

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

Evaluate $\lim_{x \to \infty}xe^{-x}$, using the rewriting $xe^{-x} = \frac{x}{e^x}$.

The limit $\lim_{x \to \pi}\frac{\cos(x)}{x}$ equals $-\frac{1}{\pi}$ by direct substitution, because the quotient is continuous at $x = \pi$.