Mathfolis

L'Hôpital's Rule

Unit 4 · Contextual Applications of Differentiation

L'Hôpital's Rule

When a limit yields an indeterminate form 0/0 or ∞/∞, differentiate the numerator and denominator separately (not the quotient rule) and retry the limit. Apply repeatedly if needed, but always verify the indeterminate form each time.

L'Hôpital's Rule

If lim f(x)/g(x) → 0/0 or ∞/∞, then
lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

Converting Other Indeterminate Forms

0 · ∞: rewrite as a fraction
f⋅g=f1/gorg1/ff \cdot g = \frac{f}{1/g} \quad\text{or}\quad \frac{g}{1/f}
1∞,00,∞01^∞, 0^0, ∞^0: take ln first
lim⁡eh(x)ln⁡g(x)  ⇒  apply L’Hoˆpital to exponent\lim e^{h(x)\ln g(x)} \;\Rightarrow\; \text{apply L'Hôpital to exponent}
Caution: Always verify the limit produces 0/0 or ∞/∞ before applying L'Hôpital's Rule — applying it to a non-indeterminate form gives the wrong answer.
AP Tip: You may apply L'Hôpital's Rule repeatedly, but re-check the indeterminate form each time before differentiating again.
Type 1

Basic 0/0 Form

Verify the form, differentiate numerator and denominator separately, then take the limit.

Example 1
Evaluate the limit.
lim⁡x→0sin⁡xx\lim_{x \to 0} \frac{\sin x}{x}
Example 2
Evaluate the limit (may require two applications).
lim⁡x→0ex−1−xx2\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}

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Type 2

∞/∞ Form

When both numerator and denominator diverge to infinity, L'Hôpital's Rule still applies.

Example 3
Evaluate the limit.
lim⁡x→∞ln⁡xx\lim_{x \to \infty} \frac{\ln x}{x}
Example 4
Evaluate the limit.
lim⁡x→∞x2ex\lim_{x \to \infty} \frac{x^2}{e^x}

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Type 3

Other Indeterminate Forms

Convert 0·∞ to a fraction, or use ln to handle exponential forms like 1∞,00,∞01^∞, 0^0, ∞^0.

Example 5
Evaluate the limit.
lim⁡x→0+xln⁡x\lim_{x \to 0^+} x \ln x
Example 6
Evaluate the limit.
lim⁡x→0+xx\lim_{x \to 0^+} x^x

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