Question

What is the limit of `lnx` as x approaches `0`?

Answer 1

`lim_(xrarr0)lnx=-oo`, ie the limit does not exists as it diverges to `-oo`

Explanation:

You may not be familiar with the characteristics of `ln x` but you should be familiar with the characteristics of the inverse function, the exponential `e^x`:

Let `y=lnx=> x = e^y`, so as `xrarr0 => e^yrarr0`

You should be aware that `e^y>0 AA y in RR`,but `e^yrarr0` as `xrarr-oo`.

The graph of `f(x)=e^x` should help illustrate this:
graph{e^x [-10, 10, -5, 5]}

so if we want `e^yrarr0=>yrarr-oo`

Therefore we can conclude that `lim_(xrarr0)lnx=-oo`, ie the limit does not exist as diverges to `-oo`

The graph of `f(x)=lnx` should help illustrate this:
graph{lnx [-10, 10, -5, 5]}