What is the limit as x approaches infinity of #ln(x)#?
To see this, we'll use:
We will look at intervals of the form:
Since this integral is
The answer is
You can prove it by reductio ad absurdum.
You know that if
You also know that
For any strictly increasing function, f, if
Let M be an arbitrary positive number.
Since (as others have shown),
if and only if
Since the numbers themselves increase without bound, we have shown that by making x large enough, we may make
Thus, the limit is infinite as x goes to