Nature of an improper integral

Question

I want to study the convergence of this integral at 0: $$ \int_0^{1}\frac{e^{\frac 1 t}}{\sqrt{t(1+t^2)}}\;dt. $$

Answer 1

$$\int_{\varepsilon}^{1}\frac{e^{1/t}}{\sqrt{t(1+t^2)}}\,dt = \int_{1}^{1/\varepsilon}\frac{e^t}{\sqrt{t(1+t^2)}}\,dt$$ and the exponential function grows too fast to make the RHS converging as $\varepsilon\to 0^+$.

For instance, since $t+t^3<e^{6t/5}$ on $[1,+\infty)$,

$$\int_{1}^{1/\varepsilon}\frac{e^t}{\sqrt{t(1+t^2)}}\,dt \geq \int_{1}^{1/\varepsilon}e^{2t/5}\,dt = \frac{5}{2}\left(\exp\frac{2}{5\varepsilon}-\exp\frac{2}{5}\right).$$