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Limits & Continuity · Special Limits

L'Hopital's Rule

If 00 or ±∞±∞:lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\text{If } \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}: \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

L'Hopital's Rule: When a limit gives an indeterminate form 0/0 or ∞/∞, the limit equals the ratio of the derivatives (if that limit exists).

Conditions. The limit must produce 0/0 or ±∞/±∞. g'(x) ≠ 0 near a. The derivative limit must exist (or be ±∞).

Worked examples

Find lim(x→0) sin(x)/x.
  1. Direct substitution gives 0/0 -indeterminate
  2. Apply L'Hopital: lim(x→0) cos(x)/1
  3. = cos(0)/1 = 1

Answer: 1

Find lim(x→∞) ln(x)/x.
  1. Direct substitution gives ∞/∞ -indeterminate
  2. Apply L'Hopital: lim(x→∞) (1/x)/1 = lim(x→∞) 1/x
  3. = 0

Answer: 0

Find lim(x→0) (eˣ - 1)/x.
  1. Direct substitution gives 0/0
  2. Apply L'Hopital: lim(x→0) eˣ/1 = e⁰

Answer: 1

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