Navigation: Wiki Home > Skills Index > Indeterminate Forms
This concept page covers indeterminate forms and how L'Hôpital's Rule helps evaluate them.
An indeterminate form is a limit expression that doesn't immediately reveal the answer. The form doesn't determine the limit - further analysis is needed.
| Form | Example |
|---|---|
| $\frac{0}{0}$ | $\lim_{x \to 0} \frac{\sin x}{x}$ |
| $\frac{\infty}{\infty}$ | $\lim_{x \to \infty} \frac{e^x}{x^2}$ |
| $0 \cdot \infty$ | $\lim_{x \to 0^+} x \ln x$ |
| $\infty - \infty$ | $\lim_{x \to 0^+} \left(\frac{1}{x} - \frac{1}{\sin x}\right)$ |
| $0^0$ | $\lim_{x \to 0^+} x^x$ |
| $1^\infty$ | $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x$ |
| $\infty^0$ | $\lim_{x \to \infty} x^{1/x}$ |
If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, and if $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ exists (or is $\pm\infty$), then:
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$
Rewrite as $\frac{f}{1/g}$ or $\frac{g}{1/f}$ to get $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
Find a common denominator or factor to combine into a single fraction.
Let $y = f(x)^{g(x)}$, then $\ln y = g(x) \ln f(x)$. Find $\lim \ln y$, then $\lim y = e^{\lim \ln y}$.
$$\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1$$
$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x$$ Let $y = \left(1 + \frac{1}{x}\right)^x$. Then $\ln y = x \ln\left(1 + \frac{1}{x}\right)$. $$\lim_{x \to \infty} \ln y = \lim_{x \to \infty} \frac{\ln(1 + 1/x)}{1/x} = 1$$ Therefore, $\lim y = e^1 = e$.