How to compute $\lim_{x \to \infty} \sqrt{x}(e^{-1/x}-1)$? I need to compute $$\lim_{x \to \infty} \sqrt{x}(e^{-1/x}-1)$$ using L'Hôpital's rule. Please hint me how to do it.
Thanks!

Let $\dfrac1x=h$
$$\lim_{h\to0^+}\frac{e^{-h}-1}{\sqrt h}$$
$$=\lim_{h\to0^+}\frac{e^h-1}h\cdot\frac{\lim_{h\to0^+}\sqrt h}{-\lim_{h\to0^+}e^h}=\cdots$$
