Limits involving exponential functions

[ahmadmz]

Homework Statement

I was studying L'Hopital's rule and how to deal with indeterminate forms of the type 0^0.

It's not clear to me how lim e^f(x) = e^lim f(x).

In wikipedia http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule" (under other indeterminate forms)
it says "It is valid to move the limit inside the exponential function because the exponential function is continuous".

But that would mean lim(x->a) e^f(x) = e^f(a).

Homework Equations

The Attempt at a Solution

 

[rock.freak667]

$$\lim_{x \rightarrow a}e^{f(x)} = e^{\lim_{x \rightarrow a} f(x)}$$


But for lim_x->af(x)=f(a) would only occur if f(x) is continuous at x=a

 

[Sankaku]

Remember that f(x) may have issues at a, as mentioned above, even when it is part of e^f(x). You can move the limit inside the exponential, because the exponential itself doesn't have problem spots ("is continuous everywhere"), so it is only the f(x) inside that you have to deal with regarding the limit.

 

[ahmadmz]

Yea it makes sense, but is there some way to show this as a proof?
For instance the basic limit properties such as lim(f + g) = lim f + lim g are proved.

 

[slider142]

It should be straightforward to show from the limit definition of "continuous function" that if g is continuous at L and $$\lim_{x\rightarrow a} f(x) = L$$, then $$\lim_{x\rightarrow a} g(f(x)) = g(L)$$.