Evaluate the limit.

First, we write

From this exercise (Section 7.11, Exercise #4) we know that as we have

Therefore, as ,

So, we then have

(Here we could say that since the exponential is a continuous function we can bring the limit inside and so this becomes . I’m not sure we know we can pass limits through continuous functions like that, so we continue on with expanding the exponential as in previous exercises.)

Since as we take the expansion of as ,

Therefore,

Below, I tried to prove that we can bring the limit inside since the exponential is a continuous function:

If is continuous in an interval I and I, then

Since the limits of g at right member of equation exist.

We can do , then for all , we have a such that

|g(x)-L| |f(g(x))-f(L)|\deltax \to -\inftyx \to \infty|lim_{x\to p}g(x)|= \infty\epsilon >0\delta

f(y)|lim_{x\to p} y|= \infty|lim_{x\to \infty}g(x)|= \infty$, it’s analogous. Therefore, in any cases above, you only have to investigate the limit of g.

Ops, I don’t know how to use latex here. Nevertheless, I posted that same aswer in this link: http://math.stackexchange.com/questions/1313755/why-is-lim-limits-x-to-infty-e-lny-e-lim-limits-x-to-infty-ln/1839666#1839666