# Question #c6e05

Jan 20, 2018

The limit does not exist.

#### Explanation:

So we have ${\lim}_{x \to \infty} \arccos \frac{{e}^{x}}{x}$

Before we do anything, we ask ourselves: What is the domain of this function?

As long as $x \ne 0$ and ${e}^{x} - 1 \le 1$, then our function is valid.
This is because first, we cannot divide by zero, and second, cosine of an angle can only range from $- 1$ to $1$
We solve our inequality:${e}^{x} - 1 \le 1$
${e}^{x} \le 2$
$x \le \ln 2$
Note that $\ln 2 \approx 0.693$

So our domain is $\left[- \infty , 0\right) \cup \left(0 , \ln 2\right)$

If we get outside this domain, the $y$ value will always be undefined.

Therefore, the limit we are looking for does not exist.