What Continuity Property of e Ensures Limits Hold True?

[brunokabahizi]

Homework Statement

If the lim_x→b f(x)=c, then lim_x→b e^x= e^c. What property of the function g(x)=e^x allows this fact?

The Attempt at a Solution

Is it just because e is a constant?

 

[AdrianZ]

No. e is a constant, but e^x is not. and probably you meant $lim_{x→b}e^{f(x)}=e^c$. well, I think the answer is continuity.

 

[brunokabahizi]

Yes, you're right. I did mean lim_x→b e^f(x)= e^c. But how does continuity allow it? Excuse me for being incompetent.

 

[HallsofIvy]

Did you consider looking up "continuous" in your text?

That property pretty much is the definition of "continuous":

The function f(x) is continuous at x= a if and only if
(1) f(a) exists.
(2) $\lim_{x\to a} f(x)$ exists.
(3) $\lim_{x\to a} f(x)= f(a)$.

Since (3) pretty much implies the left and right sides exist, of only that is given as the definition- but that's really "shorthand".

 

[AdrianZ]

well, I meant that the function needs to be continuous at the point x=c. There's a theorem in Calculus that talks about the composition of functions. the theorem states that if $lim_{x→b}f(x)=c$ and $lim_{x→a}g(x)=b$ then $lim_{x→a} fog(x)=c$ is NOT true in general, but this law holds if f is continuous at x=b.
That's why I said continuity is the key. The theorem seems to be easy to be proved though.