Question

Lim x approaches 0 cot2xsin6x

Answer 1

plugging in x = 0 gives cot 2x as indeterminate and sin 6x = 0 so that doesn't help at all
maybe a conversion to a Taylor series would help

Answer 2

split it out into sin and cos and use these known limits
\[\lim_{x \rightarrow 0} \frac{\sin(ax)}{\sin(bx)} = \frac{a}{b}; \ \lim_{x \rightarrow a} \cos(x) = \cos (a)\]
the first you can reverse into from:
\[\lim_{x \rightarrow 0} \frac{\sin(ax)}{x} = a; \ \lim_{x \rightarrow 0} \frac{\sin(bx)}{x} = b\]