§3.2 Using derivatives to evaluate limits

Introduction

Using derivatives to evaluate indeterminate limits of the form

Limits involving

Let $h(x) = \frac{x^5 + x - 2}{x^2 - 1}$. The domain of $h$ is

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

Evaluate $\lim_{x \to \frac{\pi}{2}^-}\frac{\tan(x)}{x-\frac{\pi}{2}}$, noting the form carefully.

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

Evaluate $\lim_{x \to 0}\frac{\sin(x) - x}{\cos(2x)-1}$, which needs L'Hopital twice.

Graphs show $f(2) = 0$, $f'(2) = \frac{1}{2}$, $g(2) = 0$, and $g'(2) = 4$. Compute $\lim_{x \to 2}\frac{f(x)}{g(x)}$ as a decimal.

Graphs show $p(2) = q(2) = 1.5$ with $q'(2) = 0$. The limit $\lim_{x \to 2}\frac{p(x)}{q(x)}$

Graphs show $r(2) = r'(2) = s(2) = s'(2) = 0$ with $r''(2) \ne 0$ and $s''(2) \ne 0$, where $r$ is concave up and $s$ is concave down near $2$. What can be said about $\lim_{x \to 2}\frac{r(x)}{s(x)}$?

For $q(x) = \frac{3x^2 - 4x + 5}{7x^2 + 9x - 10}$, compute $\lim_{x \to \infty}q(x)$ as a decimal rounded to three places.

Evaluate $\lim_{x \to 0^+}\frac{\ln(x)}{\frac{1}{x}}$, which has the form $\frac{-\infty}{\infty}$.

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 $x = \pi$.