Question

Quick question about L'Hospital's rule: Do you just take the derivative once, or keep taking it the derivative until you get a non-indeterminate form?

Answer 1

Take for instance the following limit:
\[L=\lim_{x\to\infty}\frac{x^2}{e^x}\]
Directly substituting yields the indeterminate form \(\dfrac{\infty}{\infty}\).

Suppose you apply L'Hopital's rule once:
\[L=\lim_{x\to\infty}\frac{2x}{e^x}=\frac{\infty}{\infty}\]
Given another indeterminate form, you're allowed to apply the rule again:
\[L=\lim_{x\to\infty}\frac{2}{e^x}=0\]
So, to answer your question, yes: you may apply L'Hopital's rule as many times as necessary.

However, there are some limits for which repeated applications of L'Hopital's rule won't work. For example,
\[\lim_{x\to\infty}\frac{\sqrt{x+2}}{\sqrt{x-1}}=\frac{\infty}{\infty}\\
\lim_{x\to\infty}\frac{\frac{1}{2\sqrt{x+2}}}{\frac{1}{2\sqrt{x-1}}}=\lim_{x\to\infty}\frac{\sqrt{x-1}}{\sqrt{x+2}}\]
When this kind of situation arise, you'll have to look for other methods to compute the limit.

Answer 2

Thanks so much, I get it now

Answer 3

You're welcome