How do you know when to use L'hospital's rule twice?

Jun 14, 2018

As soon as you try substitution and see you're in the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, you may use l'hospitals. Let's try an example!

$L = {\lim}_{x \to 0} \frac{{e}^{x} - x - 1}{x} ^ 2$

Try substitution on this and you will get $\frac{0}{0}$.

$L = {\lim}_{x \to 0} \frac{{e}^{x} - 1}{2 x}$

Now try substitution again to get $L = \frac{{e}^{0} - 1}{2 \left(0\right)} = \frac{0}{0}$. So we may indeed apply l'hospitals once more.

$L = {\lim}_{x \to 0} \frac{{e}^{x}}{2}$

Now we can evaluate directly and see that the limit is $\frac{1}{2}$.

However there will be times when you may not use l'hospitals more than once. Take the following.

$L = {\lim}_{x \to 0} \frac{{e}^{x} - 1}{x} ^ 2$

Try substitution and it'll yield $\frac{0}{0}$.

$L = {\lim}_{x \to 0} {e}^{x} / \left(2 x\right)$

Now try substitution and you will get $\frac{1}{0}$, which is undefined, therefore the limit DNE. Recall that l'hospitals may only be used when you're of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

Hopefully this helps!