# Properties of the constant e

## Homework Statement

If the limx→b f(x)=c, then limx→b ex= ec. What property of the function g(x)=ex allows this fact?

## The Attempt at a Solution

Is it just because e is a constant?

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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.

Yes, you're right. I did mean limx→b ef(x)= ec. But how does continuity allow it? Excuse me for being incompetent.

HallsofIvy
Homework Helper
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".

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.