Question

How do you find the limit of `[1/ln(x)] - [1/x-1]` as x approaches 1?

Answer 1

The limit does not exist.

Explanation:

As `xrarr1`, the second term goes to `0`.

`lim_(xrarr1)(1/x-1) = 1/1-1=0`

So the limit depends entirely on the limit of the first term.

As `xrarr1`, the natural logarithm goes to `0`,

`lim_(xrarr1)lnx = 0`

Therefore, `1/lnx` is either increasing or decreasing without bound (going to `oo` or to `-oo`) depending on whether `x` is to the right or left of `1`.

For `0 < x < 1`, we have `lnx < 0` so, `lim_(xrarr1^-)1/lnx = -oo`.

On the other hand, for `1 < 1`, we have `lnx > 0`, so `lim_(xrarr1^+)1/lnx = oo`.

The two-sided limit does not exist.