How to find the limit $lim_{t \to0+}\frac{e^{-1/t}}{\sqrt t}$

Question

How can I calculate $lim_{t \to0+}\frac{e^{-1/t}}{\sqrt t}$ ? I know that it goes to 0, but want to know how to get it.

Answer 1

You could use l'Hôpital's rule or let $x=\tfrac{1}{t}$, then $x \to +\infty$ as $t \to 0^+$ and the limit becomes: $$\lim_{t\to 0^+}\frac{e^{-1/t}}{\sqrt{t}}=\lim_{x \to \infty}\sqrt{x}e^{-x}=\lim_{x \to \infty}\frac{\sqrt{x}}{e^x}=\ldots$$ Now you might know that the exponential dominates any power function $x^a$ (with $a>0)$.

Answer 2

$$\lim_{t\rightarrow 0^+}\frac{e^{\frac{-1}{t}}}{\sqrt{t}}=\lim_{u\rightarrow\infty}\sqrt{u}e^{-u}=\lim_{u\rightarrow\infty}\frac{\sqrt{u}}{e^{u}}$$ You can then use l'Hôpital to get $$\lim_{u\rightarrow\infty}\frac{\sqrt{u}}{e^u}=\lim_{u\rightarrow\infty}\frac{1}{2\sqrt{u}e^u}=0$$

Answer 3

$$\lim_{t \to0+}\frac{e^{-1/t}}{\sqrt t}=\lim_{x \to \infty}e^{-x} \sqrt{x} = 0$$