What is the limit as x approaches infinity of ##ln(x)##?

Infinity.

Explanation:

As you can see from these 2 graphs of ##ln(x)##:

It seems that as ##x## continues to move in the positive direction, the value of ##y## keeps increasing, although slowly. You could also try plugging in larger and larger numbers into the ##ln(x)## function and see that it continues to result in larger and larger answers. Because the ##y## value continues increasing to ##oo##, the limit of ##ln(x)## as ##x## moves very far to the right (such as near ##oo##), is also ##oo##:

##lim_(x->oo) ln(x) = oo##

Explanation 2:

You can also look at the derivative of ##ln x## which is ##1/x##. Since this derivative is positive, it means the function is constantly increasing even if it is very small. In terms of basic functions, ##ln x## has the slowest growth. The only way to get slower growing functions is to have composite functions such as ##ln(ln x)##.