Proof for {lim of exp = exp of lim}

[DocZaius]

I don't know how to Google appropriately for this, since the kind of keywords I use present me with search results that try to define the exponential function using limits instead of what I am trying to ask:

What does the proof look like for the following (assuming f(x) is "nice"). Any sites that can show this? Thanks.

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

 

[blue_leaf77]

Consider expanding the exponential into power series then use the distributive property of limit.

 

[fresh_42]

You can search for "continuity of the exponential function", e.g. http://www.bookofproofs.org/branches/continuity-of-exponential-function/proof/

 

[dextercioby]

Consider expanding the exponential into power series then use the distributive property of limit.

That is a circular proof. In order to Taylor/MacLaurin expand a function, you need to have proven its continuity.

 

[blue_leaf77]

You can search for "continuity of the exponential function", e.g. http://www.bookofproofs.org/branches/continuity-of-exponential-function/proof/

Well I interpret the OP's wording of "assuming f(x) is "nice"" as to be having all orders of derivative.

 

[Svein]

Hm. I wonder...

Define a function Φ(x) as Φ(x)=0 for x≤0; $\phi(x)=e^{-\frac{1}{x^{2}}}$ for x>0. Now Φ(x) is "nice" as it is C^∞, even at x=0.

 

[fresh_42]

Well I interpret the OP's wording of "assuming f(x) is "nice"" as to be having all orders of derivative.

Maybe, but is smoothness of $f$ necessary? Without considering details it looks like pointwise convergence at $c$ should be enough.

 

[geoffrey159]

It seems that it is just an application of the rule of composition of limits (if $f$ nice means that it has a finite limit in $c$).

The rule states that if $h(x) \to b$ as $x\to a$, and if $g(x) \to \ell$ as $x \to b$, then $g(h(x)) \to \ell$ as $x\to a$.
If you add $g$ continuous (like $x\to\exp(x)$) then $\ell = g(b)$

 

[WWGD]

I think , like Fresh_42 said, that continuity is one of the simplest ways of going about it, continuity implies sequential continuity (though the converse is not true)