If you have access to Mathematica, I find it very useful to look at the graph. Although this problem is very solvable on paper as well.
Direct substitution won't work, because it's indeterminate. One method is to let t=x,t=y and find the limit of a single variable, t. This makes the limit (t)/(5t) = 1/5. This is effectively approaching the limit along the line x=y.
To try a different line, let t=-x,t=y (the line y=-x). From this approach, the limit is (-5t)/(t) = -5. Since these values are different, the function approaches a different value from different directions, and the limit does not exist. This method will only work to prove a limit's non-existence. You cannot prove that a limit exists with this methods, only disprove it.
You can basically use direct substitution, just keep in mind that the sin(anything) must be within [-1,1]. As the values approach zero, the limit's upper and lower bounds also go to zero, so you can effectively use the squeeze/sandwich theorem, and then (0)(-1) <= answer <= (0)(1) and 0 <= answer <= 0