Is $e^{\sqrt{2}}\gt 3$ or $e^{\sqrt{2}}\lt 3$

Question

$e^{\sqrt{2}}\gt 3$ or $e^{\sqrt{2}}\lt 3$ which one holds true $?$ I know that $2\lt e \lt 3$ and $\sqrt{2}\gt 1$. Little help on how to use them to find the right inequality.

Thanks for any help.

Answer 1

Using the series of $\exp$ we have $e^{\sqrt{2}} \ge 1 + \sqrt{2} + \frac{2}{2!} > 3$.

Answer 2

If you can use that $e>2.7$ and $\sqrt 2> 1.4$, then $$e^{\sqrt 2} > 2.7^{1.4} = (1+1.7)^{1.4} > 1+1.4\cdot 1.7 = 3.38$$ by Bernoulli's inequality.