How to show the sequence $x_n = (1 + \frac{x}{n})^{n}$ is bounded above by $e^x$?

The result is not true for $x<0$. A better approach for $x >0$: If $x >0$ the $(1+\frac x n)^{n}=e^{n\log (1+x/n)}\leq e^{n\frac x n}=e^{x}$ where I have used the inequality $\log(1+y) \leq y$ for all $y >0$.

Hint: If $t>0$ then $$\log(1+t)=\int_1^{1+t}\frac{ds}s<\int_1^{1+t}ds=t.$$