If going to the limit results in a 0/0 or infinity/innfinity indeterminate fraction, then take the ratio of the derivatives of the numerator and denominator to get the limit. That is called L'Hopital's rule.
For the question
Lim (sin (2x)) / (sin (3x))
the answer is [2 (cos(0)]/[3 cos(0)] = 2/3
Remember that sin x is very close to x when x is small. this tells you right away that it approaches 2x/3x = 2/3
For a website that should help, Google "L'Hopital's rule".
One good site is
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