when evaluating a limit, the forms $\frac{0}{0},$ $\infty \text{/} \infty,$ $0 \cdot \infty,$ $\infty - \infty,$ $0^0,$ $\infty^0,$ and $1^{\infty}$ are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is

Calculus Volume 1 · 4.8 L’Hôpital’s Rule

L’Hôpital’s Rule: practice problems

28 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 357
For the following exercises, evaluate the limit.
Evaluate the limit $lim_{x \to \infty} \frac{e ^{x}}{x ^{k}} .$
Sign up free to have it checked
Exercises 359
For the following exercises, evaluate the limit.
Evaluate the limit $lim_{x \to a} \frac{x - a}{x ^{2} - a ^{2}}, a \neq = 0$ .
Sign up free to have it checked
Exercises 361
For the following exercises, evaluate the limit.
Evaluate the limit $lim_{x \to a} \frac{x - a}{x ^{n} - a ^{n}}, a \neq = 0$ .
Sign up free to have it checked
Exercises 363
For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.
$lim_{x \to \infty} x^{1 / x}$
Sign up free to have it checked
Exercises 365
For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.
$lim_{x \to 0} \frac{x ^{2}}{1 / x}$
Sign up free to have it checked
Exercises 367
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$
Sign up free to have it checked
Exercises 369
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 0} \frac{( 1 + x ) ^{- 2} - 1}{x}$
Sign up free to have it checked
Exercises 371
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to π} \frac{x - π}{s i n x}$
Sign up free to have it checked
Exercises 373
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 0} \frac{( 1 + x ) ^{n} - 1}{x}$
Sign up free to have it checked
Exercises 374
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 0} \frac{( 1 + x ) ^{n} - 1 - n x}{x ^{2}}$
Sign up free to have it checked
Exercises 375
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 0} \frac{s i n x - t a n x}{x ^{3}}$
Sign up free to have it checked
Exercises 377
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 0} \frac{e ^{x} - x - 1}{x ^{2}}$
Sign up free to have it checked
Exercises 379
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 1} \frac{x - 1}{l n x}$
Sign up free to have it checked
Exercises 381
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 1} \frac{x - 3 x}{x - 1}$
Sign up free to have it checked
Exercises 383
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to \infty} x sin (\frac{1}{x})$
Sign up free to have it checked
Exercises 385
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 0^{+}} x ln (x^{4})$
Sign up free to have it checked
Exercises 387
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to \infty} x^{2} e^{- x}$
Sign up free to have it checked
Exercises 389
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 0} \frac{1 + 1 / x}{1 - 1 / x}$
Sign up free to have it checked
Exercises 391
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to \infty} x e^{1 / x}$
Sign up free to have it checked
Exercises 393
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to 0^{+}} x^{1 / x}$
Sign up free to have it checked
Exercises 395
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
$lim_{x \to \infty} (1 - \frac{1}{x})^{x}$
Sign up free to have it checked
Exercises 397
For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.
[T] $lim_{x \to 0} x sin (\frac{1}{x})$
Sign up free to have it checked
Exercises 399
For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.
[T] $lim_{x \to 1} \frac{e ^{(x - 1)} - 1}{x - 1}$
Sign up free to have it checked
Exercises 401
For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.
[T] $lim_{x \to π} \frac{1 + c o s x}{s i n x}$
Sign up free to have it checked
Exercises 403
For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.
[T] $lim_{x \to 0^{+}} tan (x^{x})$
Sign up free to have it checked
Exercises 405
For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.
[T] $lim_{x \to 0} \frac{e ^{x} - e ^{- x}}{x}$
Sign up free to have it checked
Review Exercises 539
Evaluate the following limits.
$lim_{x \to \infty} \frac{3 x x ^{2} + 1}{x ^{4} - 1}$
Sign up free to have it checked
Review Exercises 541
Evaluate the following limits.
$lim_{x \to 1} \frac{x - 1}{s i n ( π x )}$
Sign up free to have it checked

Key terms · 2

indeterminate forms: when evaluating a limit, the forms $\frac{0}{0},$ $\infty \text{/} \infty,$ $0 \cdot \infty,$ $\infty - \infty,$ $0^0,$ $\infty^0,$ and $1^{\infty}$ are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is
L’Hôpital’s rule: if $f$ and $g$ are differentiable functions over an interval $a,$ except possibly at $a,$ and $\lim_{x \to a} f (x) = 0 = \lim_{x \to a} g (x)$ or $\lim_{x \to a} f (x)$ and $\lim_{x \to a} g (x)$ are infinite, then $\lim_{x \to a} \frac{f (x)}{g (x)} = \lim_{x \to a} \frac{f' (x)}{g' (x)},$ assuming the limit on the right exists or is $\infty$ or $\text{-} \infty$

Learn 4.8 on the whiteboard

The tutor teaches the whole section, drawing it out step by step and answering your questions as you go.

Read 4.8 in the textbook · This section on openstax.org

L’Hôpital’s Rule: practice problems

28 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 357

For the following exercises, evaluate the limit.

Evaluate the limit $lim_{x \to \infty} \frac{e ^{x}}{x ^{k}} .$

Exercises 359

For the following exercises, evaluate the limit.

Evaluate the limit $lim_{x \to a} \frac{x - a}{x ^{2} - a ^{2}}, a \neq = 0$ .

Exercises 361

For the following exercises, evaluate the limit.

Evaluate the limit $lim_{x \to a} \frac{x - a}{x ^{n} - a ^{n}}, a \neq = 0$ .

Exercises 363

For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.

$lim_{x \to \infty} x^{1 / x}$

Exercises 365

$lim_{x \to 0} \frac{x ^{2}}{1 / x}$

Exercises 367