I was trying to solve an improper integral and I had to evaluate the following expression with the upper limit as infinite (the limit is 0) and the lower limit as 0 (the limit I am asking here)

Here is the limit:

$$ lim_ {x a 0} (x ln (e ^ {x} -1) -x ^ {2}) $$

I introduced the expression in an online limit solver.

The L & # 39; Hopital rule can be used when the limit is in an undetermined form, and when there is the limit in the numerator and the limit in the denominator. The limit of $$ frac {1} {x} $$

It does not exist when x approaches 0. (And I think the limit of the logarithm in the numerator does not exist when it approaches 0, unless only from the right? Someone confirm this please)

Is it safe to assume that the boundary solver assumes that x approaches 0 from the right?

Does this mean that the limit only exists when x is approached from the right?

And if this limit does not exist unless it is from the right, is it safe to change the previous limit so that it approaches from the right to solve this integral?