Indeterminate Forms

Understanding Indeterminate Forms

When evaluating $\lim_{x\to a} \frac{f(x)}{g(x)}$ through direct substitution, you sometimes encounter $\frac{0}{0}$ or $\frac{\infty}{\infty}$. These are indeterminate forms—the limit could equal any value or fail to exist entirely, depending on the specific functions involved. Elementary techniques like factoring or rationalizing resolve some cases, but many require a more systematic approach.

L'Hopital's Rule

If $\lim_{x\to a} \frac{f(x)}{g(x)}$ produces $\frac{0}{0}$ or $\frac{\infty}{\infty}$, and both $f$ and $g$ are differentiable near $a$ with $g'(x) \neq 0$ nearby, then:

$\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)}$

Key point: Differentiate the numerator and denominator separately—this is not the quotient rule. The rule applies to limits as $x \to \pm\infty$ as well.

Verification and Reapplication

Always verify that substitution yields $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before applying the rule; using it on determinate forms produces incorrect answers. After differentiating once, check the new limit. If it's still indeterminate, reapply the rule. Stop once direct substitution yields a determinate value.

Example: $\lim_{x\to 0} \dfrac{\sin x}{x}$

Direct substitution: $\frac{0}{0}$ ✓ Apply the rule: $\lim_{x\to 0} \frac{\sin x}{x} = \lim_{x\to 0} \frac{\cos x}{1} = 1$

Example: $\lim_{x\to \infty} \dfrac{x^2}{e^x}$

Direct substitution: $\frac{\infty}{\infty}$ ✓ First application: $\lim_{x\to \infty} \frac{2x}{e^x} \quad \text{(still } \frac{\infty}{\infty}\text{)}$

Second application: $\lim_{x\to \infty} \frac{2}{e^x} = 0$

Common Pitfalls

Avoid applying L'Hopital's Rule to non-indeterminate forms like $\frac{5}{0}$. Never substitute the quotient rule inside the rule itself—differentiate $f$ and $g$ independently. Other indeterminate forms ($0 \cdot \infty$, $\infty - \infty$, $1^\infty$) must first be algebraically rewritten as $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before proceeding.