Can someone show me how:
$\displaystyle \lim_{t\to 0} (1+t)^{\frac{1}{t}}=e$
Say that this limit exists and it equals A.
Then $\displaystyle =\frac{1}{t}\ln(1+t)=\ln(A)$
The LHS is also $\displaystyle \frac{\ln(1+t)}{t}$ and when evaluating at t=0 you get 0/0, which is indeterminate and L'Hopitals can be used.
So $\displaystyle \ln(A)=\frac{\frac{1}{1+t}}{1}$
Try the limit again, and use algebra to solve for A again.
edit: This is in the precalc section, so forgive me for using calculus.
Alternate solution:
recall that $\displaystyle e^{\ln X} = X$ for any $\displaystyle X > 0$.
so that:
$\displaystyle \lim_{t \to 0}(1 + t)^{\frac 1t} = \lim_{t \to 0}\text{exp} \left( \frac 1t \ln (1 + t) \right)$
$\displaystyle = \text{exp} \left( \lim_{t \to 0} \frac {\ln (1 + t)}t \right)$
now since $\displaystyle \lim_{t \to 0} \frac {\ln (1 + t)}t = 1$ (By L'Hopital's rule, or power series or ...)
we have that the limit is $\displaystyle \text{exp}(1) = e$
This is very similar to Jameson's method, both of these methods you should consider when taking the limit of an expression with the variable in the power.
That is, let $\displaystyle A $ and $\displaystyle B$ be expressions in a single variable in which the variable appears, then we can usually find various limits that give indeterminate forms by doing the following:
The method Jameson used:
Say $\displaystyle y = A^B$
Then $\displaystyle \ln y = B \ln A$
so that $\displaystyle \lim \ln y = \lim B \ln A$
and so $\displaystyle \lim y = e^{\lim B \ln A}$ by taking the anti-log of both sides.
The method I used:
simply write $\displaystyle y = A^B$ as $\displaystyle e^{\ln A^B} = e^{B \ln A}$, and take the limit of both sides.
note that $\displaystyle \lim e^X = e^{\lim X}$
the rigorous theory might get a bit complicated, but it is somewhat similar to the fact that we can factor constants out side of limits. since the limit deals with X and not e, taking the limit of e to some power really only affects the power, since anything that is not a variable in X is not affected by the limit.
EDIT: Chop Suey was a bit more rigorous .