# How to show that limit $\lim\limits_{x\to 0}e^\frac1x$ does not exist?

Show that $\lim\limits_{x\to 0}e^\frac{1}{x}$ does not exist.

I tried to show this using the epsilon delta definition but couldn't come across a contradiction.

• Hint: Consider the left and right limits. – SquirtleSquad May 2 '17 at 4:43
• As $x\to 0^+$, we have $\frac{1}{x}\to +\infty$ and in this case $e^{\frac{1}{x}}\to +\infty$ Whereas if $x\to 0^-$, we have $\frac{1}{x}\to -\infty$ and in this case $e^{\frac{1}{x}}\to 0$ – Juniven May 2 '17 at 4:44

Try this common estimate: $$e^{\frac{1}{x}}>1+\frac{1}{x}$$ and approach from the left.