Indeterminate Forms and L'Hôpital's Rule

Occasionally in mathematics, one runs across an equation with an indeterminate form as seen in . In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form. The indeterminate forms include $0^0$, $\frac{0}{0}$, $1^\infty$, $\infty - \infty$, $\frac{\infty}{\infty}$, $0 \times \infty$, and $\infty^0$.

The most common example of an indeterminate form is $\frac{0}{0}$. As $x$ approaches $0$, the ratios $\frac{x}{x^3}$, $\frac{x}{x}$, and $\frac{x^2}{x}$ go to $\infty$, $1$, and $0$, respectively. In each case, however, if the limits of the numerator and denominator are evaluated and plugged into the division operation, the resulting expression is $\frac{0}{0}$. So, roughly speaking, $\frac{0}{0}$ can be $0$, or $\infty$, or it can be $1$—in fact, it is possible to construct similar examples converging to any particular value. That is why the expression $\frac{0}{0}$ is indeterminate.

More formally, the fact that the functions $f$ and $g$ both approach $0$ as $x$ approaches some limit point $c$ is not enough information to evaluate the limit $\lim_{x\to c}\frac{f(x)}{g(x)}$. That limit could converge to any number, or diverge to infinity, or might not exist, depending on what the functions $f$ and $g$ are. For example, $\lim_{x\to 0}\frac{x}{x}$ is indeterminate.

L'Hôpital's Rule

In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit.

In its simplest form, l'Hôpital's rule states that for functions $f$ and $g$ which are differentiable, if 

$\displaystyle{\lim_{x\to c}f(x)=\lim_{x \to c}g(x) = 0 \text{ or } \pm \infty}$ 

and $\lim_{x\to c}\frac{f'(x)}{g'(x)}$ exists, and $g'(x) \neq 0$ for all $x$ in the interval containing $c$, then: 

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