Is $\int_{0^+}^{+\infty}\frac{e^{-x}}{\sqrt{x}}dx$ convergent or divergent

Question

I was reading a textbook and for the exercise part I faced this question that asked to figure if this integration is converge or divergent.

I tried to solve it with the limit comparison test and I couldn't find the proper function to compare with.

Can you tell me what do you think about this wanted function and how do you come up with that ?

I was wondering is it even a proper procedure to use limit comparison test?

$$\int_{0^+}^{+\infty}\frac{e^{-x}}{\sqrt{x}} dx$$

Answer 1

Hint:

Consider integrals over $[0,1]$ and $[1,\infty)$.

Try the limit comparison test with $e^{-x}$ as $ x \to \infty$ and $1/\sqrt{x}$ as $x \to 0$.