Further Maths - L'Hôpital's Rule

What is L’Hôpital’s rule used for?

Finding the limit of two functions divided together.

What is

\[\lim _ {x \to a} \frac{f(x)}{g(x)}\]

equivalent to?

\[\lim_{x \to a} \frac{f'(x)}{g'(x)}\]

What technique could you use for finding the value of

\[\frac{\sin(x)}{x}\]

at $x = 0$?

L’Hôpital’s rule.

What are the conditions for applying L’Hôpital’s rule for

\[\lim _ {x \to a} \frac{f(x)}{g(x)}\]

?

\[\frac{f(x)}{g(x)} = \frac{0}{0}\]

or

\[\frac{f(x)}{g(x)} = \frac{\pm \infty}{\pm \infty}\]

How could you find the limit of the product of two functions $f(x)g(x)$ when their product is undefined?

\[f(x)g(x) \equiv \frac{g(x)}{1/f(x)} \equiv \frac{f(x)}{1/g(x)}\]

and use L’Hôpital’s rule.

How could you evaluate

\[\lim _ {x \to -\infty} x e^x\]

?

\[\lim_{x \to -\infty} \frac{x}{1/e^x}\]

What do you need to consider when rewriting $f(x)g(x)$ as $\frac{f(x)}{1/g(x)}$ or $\frac{g(x)}{1/f(x)}$ in order to use L’Hôpital’s rule?

Which one has a nicer result when you integrate the top and bottom.

How could you evaluate $\lim e^{f(x)}$?

\[e^{\lim f(x)}\]

How can you tackle an indeterminate form like $1^\infty$?

Rewrite as $e^{\infty \times \ln 1}$.

How can you tell if a limit doesn’t exist?

Approach it from two different directions and see if you get different answers.

See Also