Question

Question #c6e05

Answer 1

The limit does not exist.

Explanation:

So we have `lim_(x->oo)arccos(e^x)/x`

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

As long as `x!=0` and `e^x-1<=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<=1`
`e^x<=2`
`x<=ln2`
Note that `ln2~~0.693`

So our domain is `[-oo,0)uu(0,ln2)`

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

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