L’Hôpital’s Rule & Indeterminate Forms – Rapid-Review Notes

Indeterminate Forms & L

  • Direct rule applies ONLY when the original limit gives \tfrac{0}{0} or \tfrac{\pm\infty}{\pm\infty}.

  • If valid, replace

    \displaystyle \lim_{x\to a}\frac{f(x)}{g(x)} by \displaystyle \lim_{x\to a}\frac{f'(x)}{g'(x)}

    (repeat until limit is determinate).

The 7 Classical Indeterminate Forms

  1. \tfrac{0}{0}

  2. \tfrac{\infty}{\infty}

  3. 0\cdot\infty

  4. \infty-\infty

  5. 0^0

  6. \infty^0

  7. 1^{\infty}

Converting Non-Fraction Forms

  • 0\cdot\infty: move one factor to denominator, e.g. f\,g\to \tfrac{f}{1/g}.

  • \infty-\infty: combine fractions with common denominator.

  • f(x)^{g(x)}: set L=\lim f^{g}, take \ln L = \lim g\,\ln f

When NOT to Apply L

  • If substitution gives a determinate value (e.g. \tfrac67), the rule is invalid.

  • If derivative of denominator ever =0 near limit while numerator isn’t, avoid (use algebra or other tests).

Canonical Example Limit \displaystyle \lim_{x\to0}\frac{\sin x}{x}:

  • Form \tfrac{0}{0} \rightarrow one L’Hôpital step: \lim_{x\to0}\tfrac{\cos x}{1}=1.

  • Confirms classical result.

Growth-Rate Hierarchy (useful for comparisons)

\ln x \ll x^k \ll a^{x} for large x.

Quick Checklist

  1. Substitute: identify form.

  2. If \tfrac{0}{0} or \tfrac{\infty}{\infty} \Rightarrow L

  3. If 0\cdot\infty or \infty-\infty \Rightarrow rewrite to quotient.

  4. If power-type \Rightarrow take natural log.

  5. Differentiate numerator & denominator until determinate.

  6. Always re-check that conditions stay within valid domain.