Question #c6e05

1 Answer
Jan 20, 2018

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.