Homework Statement
use delta, epsilon to prove that e^x is continuous at c = 0
Homework Equations
(a) for y>0, lim_n-> inf, y^(1/n) = 1
(b) for x < y, exp(x) < exp(y)
The Attempt at a Solution
im not sure how to approach this problem.
i have,
|exp(x) - exp(0)|= |exp(x) - 1|...
just one more question, when writing out the formal proof. how/where would i add the case for zero? since that case would need to have epsilon prime set to a different value.
the other thing i was confused about was:
i've been given an e' for a certain N'
but for my proof i have set e' as some...
Since,
|a_n - A| < ε
We have that for,
ε' > 0
there exists N such that for all n>=N,
|\sqrt{a_n} - \sqrt{A}| < ε'
so,
|\sqrt{a_n} - \sqrt{A}| < ... < \frac{|a_n - A|}{\sqrt{A}} < \frac {ε'}{\sqrt{A}} = ε
where
ε' = ε\sqrt{A}
I feel as though I'm missing some critical parts of the...
i am trying to show that for all ε > 0, there is an N such that for all n>=N i'll have,
|\sqrt{a_n} - \sqrt{A}| < ε
i'm not understanding how to relate this (below) to my epsilon.
|\sqrt{a_n} - \sqrt{A}| < \frac{|a_n - A|}{\sqrt{A}}
i still feel as if I am integrating in circles and i don't see the bigger picture.
EDIT:
just saw your most recent post. i'll try your suggestion.
thanks!