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Properties of the constant e

  • #1

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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Answers and Replies

  • #2
319
0
No. e is a constant, but e^x is not. and probably you meant [itex]lim_{x→b}e^{f(x)}=e^c[/itex]. well, I think the answer is continuity.
 
  • #3
Yes, you're right. I did mean limx→b ef(x)= ec. But how does continuity allow it? Excuse me for being incompetent.
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
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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) [itex]\lim_{x\to a} f(x)[/itex] exists.
(3) [itex]\lim_{x\to a} f(x)= f(a)[/itex].

Since (3) pretty much implies the left and right sides exist, of only that is given as the definition- but that's really "shorthand".
 
  • #5
319
0
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 [itex]lim_{x→b}f(x)=c[/itex] and [itex]lim_{x→a}g(x)=b[/itex] then [itex]lim_{x→a} fog(x)=c[/itex] 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.
 

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