Limits involving exponential functions

  • Thread starter ahmadmz
  • Start date
  • #1
ahmadmz
62
0

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" [Broken] (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


Homework Statement





Homework Equations





The Attempt at a Solution

 
Last edited by a moderator:

Answers and Replies

  • #2
rock.freak667
Homework Helper
6,223
31
[tex]\lim_{x \rightarrow a}e^{f(x)} = e^{\lim_{x \rightarrow a} f(x)}[/tex]


But for limx->af(x)=f(a) would only occur if f(x) is continuous at x=a
 
Last edited:
  • #3
Sankaku
708
11
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.
 
  • #4
ahmadmz
62
0
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.
 
  • #5
slider142
1,015
70
It should be straightforward to show from the limit definition of "continuous function" that if g is continous at L and [tex]\lim_{x\rightarrow a} f(x) = L[/tex], then [tex]\lim_{x\rightarrow a} g(f(x)) = g(L)[/tex].
 

Suggested for: Limits involving exponential functions

Replies
5
Views
337
  • Last Post
Replies
9
Views
145
Replies
8
Views
102
Replies
4
Views
601
Replies
6
Views
251
  • Last Post
Replies
16
Views
567
Replies
16
Views
430
Replies
10
Views
520
Replies
5
Views
752
  • Last Post
Replies
2
Views
370
Top