Evaluate the limit of the function (e^x -e)/(x-1), if x goes to 1?

We notice that the expression lim (e^x -e)/(x-1) is the
way to determine the derivative of the function f(x) = e^x, at the point x = 1, using
the first principle.

lim [f(x) - f(1)]/(x-1) = lim (e^x -
e)/(x-1) = f'(1)

Let f(x) =
e^x.

We'll take logarithms both
sides:

ln f(x) = ln e^x

We'll
apply the power rule of logarithms:

ln f(x) = x*ln e, but
ln e = 1

ln f(x) = x

We'll
differentiate both sides with respect to x:

f'(x)/f(x) = 1
=> f'(x) = f(x) = e^x

f'(1) = e^1
=e

Therefore, the limit of the function, if x
approaches to 1, is lim (e^x -e)/(x-1) = e.