L'Hôpital's Rule: Evaluating Indeterminate Limits

When a limit yields $0/0$ or $\infty/\infty$, you can differentiate the numerator and denominator separately to find the limit.

Core Theorem

\lim_{x \to c} \frac{f(x)}{g(x)}

is indeterminate, then:

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

*Requirement: Both

f

and

g

must be differentiable near

c

TL;DR: If you hit an indeterminate form, differentiate the top and bottom individually.

Practice Exercises

Example 01Easy

Evaluate

\lim_{x \to 0} \frac{\sin(x)}{x}

NEED A HINT?

Plug in

x=0

first to check if it's an indeterminate form.

SHOW DETAILED EXPLANATION

\frac{\sin(0)}{0} = \frac{0}{0}

. (Indeterminate)

Derivative of

\sin x

\cos x

; derivative of

x

1

\lim_{x \to 0} \frac{\cos x}{1} = \frac{1}{1} = 1

Example 02Medium

Evaluate

\lim_{x \to \infty} \frac{e^x}{x^2}

NEED A HINT?

You may need to apply L'Hôpital's Rule more than once.

SHOW DETAILED EXPLANATION

\frac{e^\infty}{\infty^2} = \frac{\infty}{\infty}

\lim_{x \to \infty} \frac{e^x}{2x} = \frac{\infty}{\infty}

again.

\lim_{x \to \infty} \frac{e^x}{2} = \infty

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