A rule for evaluating indeterminate forms.

Theorem

Suppose that f(x) → 0 and g(x) → 0 as x → a. Then

$$\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\text{lim}}\frac{f\prime \left(x\right)}{g\prime \left(x\right)}$$

if the limit on the right‐hand side exists.

For example,

$$\underset{x\to a}{\text{lim}}\frac{\sqrt{1+x}-1}{x}=\underset{x\to a}{\text{lim}}\frac{\frac{1}{2}{(1+x)}^{-1/2}}{1}=\frac{1}{2}.$$

The result also holds if f(x) → ∞ and g(x) → ∞ as x → a. Moreover, the rule applies if ‘x → a’ is replaced by ‘x →+∞’ or ‘x →−∞’.