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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
If lim⁡x→cf(x)g(x)\lim_{x \to c} \frac{f(x)}{g(x)} is indeterminate, then:
lim⁡x→cf(x)g(x)=lim⁡x→cf′(x)g′(x)\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

*Requirement: Both ff and gg must be differentiable near cc.*

Practice Exercises

Example 01Easy
Evaluate lim⁡x→0sin⁡(x)x\lim_{x \to 0} \frac{\sin(x)}{x}.
NEED A HINT?
Plug in x=0x=0 first to check if it's an indeterminate form.
SHOW DETAILED EXPLANATION

Step 1: Check Form

sin⁡(0)0=00\frac{\sin(0)}{0} = \frac{0}{0}. (Indeterminate)

Step 2: Differentiate

Derivative of sin⁡x\sin x is cos⁡x\cos x; derivative of xx is 11.

Step 3: Re-evaluate

lim⁡x→0cos⁡x1=11=1\lim_{x \to 0} \frac{\cos x}{1} = \frac{1}{1} = 1.
Example 02Medium
Evaluate lim⁡x→∞exx2\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

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

Step 2: First Pass

lim⁡x→∞ex2x=∞∞\lim_{x \to \infty} \frac{e^x}{2x} = \frac{\infty}{\infty} again.

Step 3: Second Pass

lim⁡x→∞ex2=∞\lim_{x \to \infty} \frac{e^x}{2} = \infty.
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