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.