# Why doesn't the limit of $\frac{e^{\frac1x}-1}{e^{\frac1x}+1}$ exist?

Consider $$\lim_{x \to 0} \frac{e^{\frac1x}-1}{e^{\frac1x}+1}$$

Applying L'hospital's rule for the left hand limit and right hand limit gives the same answer.

Why doesn't this limit exist?

• Is $f$ that quotient or is that quotient the result of the limit? – Filburt Apr 13 '17 at 18:04

## 2 Answers

You can't use L'Hopital for the left hand limit, because it is not of the form $\frac{\infty}{\infty}$ or any other indeterminate form.

$$\lim_{x\to0^-} e^{1/x} = 0$$

So $$\lim_{x\to 0^-}f(x)=\frac{-1}{1}$$

You can use L'Hopital to show that $\lim_{x\to 0^+} f(x)=1$.

• Oh, okay I understood my mistake. But Is L'Hospital's rule, in general valid for one sided limits? – xasthor Apr 13 '17 at 17:26
• Yes, L'Hopital can be applied to one-sided limits. You can think of it via the equality $$\lim_{x\to 0^{+}} f(x)=\lim_{y\to+\infty} f(1/y)$$ – Thomas Andrews Apr 13 '17 at 17:28

Because the left limit is $-1$ and the right is $+1$