The main formulas and rules associated with L’Hôpital’s rule are:
- Statement of L’Hôpital’s rule:
If the limit (\lim_{x \to c} \frac{f(x)}{g(x)}) takes the indefinite form (\frac{0}{0}) or (\frac{\infty}{\infty}), then:
[
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
]
provided that the limit on the right exists or is equal to (\pm \infty). - Application of the rule:
- Check that both functions (f(x)) and (g(x)) are differentiable in a neighborhood of point (c) (except, possibly, point (c) itself).
- If after the first application of L’Hôpital’s rule you still get an indefinite form, you can apply the rule again.
- Forms in which the rule applies:
- (\frac{0}{0})
- (\frac{\infty}{\infty})
- Additional cases:
If the limit is (\lim_{x \to c} f(x) = \infty) and (\lim_{x \to c} g(x) = 0), or vice versa, one can use:
[
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \quad \text{(if applicable)}
] - Example:
For the function (\lim_{x \to 0} \frac{\sin x}{x}):
- We apply L’Hôpital’s rule:
[
\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1
]