Problem: Evaluate (if it exists) the limit
Solution: This is a standard problem, we want to find a limit from the right of an expression that exists on a right reduced neighborhood of the limit point. Thus we start by substituting this point into the expression.
How did we work out those one-sided things? First, in the logarithm we substitute "1 plus", that is, numbers close to 1 and slightly greater than 1, so the outcomes are close to zero and slightly larger than zero, therefore 0+.
In the second fraction, when x is a number close to 1 satisfying
Thus we have an indeterminate difference. Combining the two parts together using algebra and cancelling somehow does not seem possible, so we try another recommended trick, using common denominator.
This looks like an indeterminate ratio, no algebraic trick seems possible, so we use the standard trick: l'Hospital's rule.
Another indeterminate ratio, it is not clear whether another l'Hospital would help, but one logarithm did disappear, so perhaps it is worth trying it again.
