Thread: Limits Help!

Use the definition of limits, find the delta which corresponds to the epsilon givn the limit:
lim ( 1+x ) ^ 1/x = e ; given the epsilon = .001
x--->0

my solns:
|(1+x)^1/x - e| < .001
so,
-.001 < (1+x)^1/x - e < .001
and,
e - .001 < (1+x)^(1/x) < e + .001
.999 < (1+x)^(1/x) < 1.0010005 <=== This is wer i got stuck on how to.. ??

Originally Posted by ^_^Engineer_Adam^_^

Use the definition of limits, find the delta which corresponds to the epsilon givn the limit:
lim ( 1+x ) ^ 1/x = e ; given the epsilon = .001
x--->0

my solns:
|(1+x)^1/x - e| < .001
so,
-.001 < (1+x)^1/x - e < .001
and,
e - .001 < (1+x)^(1/x) < e + .001
.999 < (1+x)^(1/x) < 1.0010005 <=== This is wer i got stuck on how to.. ??

We need to find a such that, for all
we have that,

The delta inequality is more conviently written as,

The second inequality can be written as,

Thus,

Here is the problem, I have no idea how to do this delta-epsilon. In fact, this limit is completely proved otherwise.
Below I have attach a beautiful hand drawn diagram that shows for the delta-epsilon statement is true.

The thin red line shows the curve
The thick blue and purple region show that it was squeezed between that area.