A simple limit problem How do you solve this limit? I know this is probably really easy.
$$
\lim_{x \to ∞} \left(f(x) = (1 / x) * e ^ x\right)
$$
 A: Try L'Hospital's Rule:
$$
\lim_{x\to \infty}\frac{f(x)}{g(x)} = \lim_{x\to \infty}\frac{f'(x)}{g'(x)}.
$$
A: Another way would be the sandwich method:
Notice that $e^{0.5x}$ is monotonic ascending, and 
et's all agree that after a certain point $n$, $e^{0.5x} >x$ for all $x > n$ (Showing this is pretty easy and I will leave it to you).
And so, if $x>n$:
$$\frac{e^x}{e^{0.5x}} < \frac{e^x}{x} < e^x $$
When $x$ tends to infinity. the left side limit approaches infinity, the right side limit approaches to infinity, so what's bound in the middle must approach infinity as well.
A: For all $x\in\mathbb{R}$, we have $e^{x/2}\ge1+x/2$. Therefore, for $x\gt0$, $e^x\ge\left(1+x/2\right)^2$, and so
$$
\begin{align}
\lim_{x\to\infty}\frac1xe^x
&\ge\lim_{x\to\infty}\frac1x\left(1+x/2\right)^2\\
&\ge\lim_{x\to\infty}x/4\\
&\to\infty
\end{align}
$$
A: Note also that by L'Hospital's Rule, $e^x$ grows quicker than any polynomial as $x$ approaches $\infty$.
A: The L'Hospital rule is applicable here, however the other answers fail to stress its limitations. Be sure to read the wikipedia article.
