Problem: Evaluate (if it exists) the limit
Solution: This is a standard problem, we want to find a limit of an expression that exists on a reduced neighborhood of the limit point. Thus we start by substituting this point into the expression and obtain "zero over zero". This is an indeterminate ratio. The standard procedure calls for using the l'Hospital rule.
Note how after every l'Hospital step we simplify the resulting expression and then do separately its "nice" parts (those that do not contribute to zero in numerator or denominator), so that they do not make the next l'Hospital unnecessarily complicated.
Three l'Hospital's were to be expected, each application "killed" one power of arctangent and logarithm. Still, is there an easier way? Definitely. Since both the numerator and the denominator are composed functions with the third power as the outer function, we can pull this third power out of the fraction and then out of the limit (it is a continuous function). Thus one l'Hospital will suffice.
Quite an improvement, isn't it? Is there any other way? Cancelling does not seem possible, nothing else comes to mind, so this second solution is probably optimal.