L'Hôpital's Rule: Evaluating Indeterminate Limits, Solving Indeterminate Limits Fast & Easy

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

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

Step 1: Check Form

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

. (Indeterminate)

Step 2: Differentiate

Derivative of

\sin x

\cos x

; derivative of

x

1

Step 3: Re-evaluate

\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

Step 1: Check Form

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

Step 2: First Pass

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

again.

Step 3: Second Pass

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

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Gary Chang

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