L'Hopital's Rule
Table of Contents
L’hopital’s rule is a method for evaluating limits of indeterminate form.
$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f’(x)}{g’(x)}$$
Where
- \( \small f(x) \) and \( \small g(x) \) are functions,
- \(\small ‘\) denotes the derivative of a function, and
- \( \small c \) is the point at which the limit is being evaluated
Sources #
Example #
Calculate the limit of \(\small f(x) = \frac{x^2 - 9}{x - 3}\) as \(\small x\) approaches \(\small 3\).
Since plugging in 3 yields 0/0 which is an indeterminate form, we can apply L’Hopital’s rule, with \( \small f(x) = x^2 - 9\) and \( \small g(x) = x - 3\).
Taking both derivatives, we get \( \small f’(x) = 2x\) and \( \small g’(x) = 1\).
Plugging into the right side of the L’Hopital’s rule, we get that
$$ \lim_{x \to 3} \frac{f(x)}{g(x)} = \lim_{x \to 3} \frac{f’(x)}{g’(x)} = \lim_{x \to 3} \frac{2x}{1} = 6$$