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.