1. 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.. ??

$/epsilon$

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.. ??

$/epsilon$
We need to find a $\delta>0$ such that, for all
$|x|<\delta$ we have that,
$\left|\left(1+x\right)^{1/x}-e\right|<.001$
The delta inequality is more conviently written as,
$-\delta
The second inequality can be written as,
$-.001<\left(1+x\right)^{1/x}-e<.001$
Thus,
$e-.001<\left(1+x\right)^{1/x}
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 $\delta=.0001$ the delta-epsilon statement is true.

The thin red line shows the curve $y=(1+x)^{1/x}$
The thick blue and purple region show that it was squeezed between that area.