FINDING THE LIMIT USING DIRECT SUBSTITUTION

lim (cos 2x)/cot 2x
x->0

I need help on this. Can you show me the steps to the answer?

lim(x->0)(cos (2x))/(cot (2x))
= lim(x->0)(cos (2x))/[(cos (2x))/(sin(2x))]
= lim(x->0)(sin(2x)) = 0

lim x->0 {(cos 2x)/(cot 2x)}
= (cos 0)/(cot 0)
= 1/(infinity)
= 0

lim (cos 2x)/cot 2x
x->0
= lim (cos 2x)/[cos 2x/sin2x]
x->0
= lim sin2x
x->0
= sin0 =0

cos 2x/cot 2x = cos 2x/(cos2x/sin 2x) = sin 2x --> 0 as x -->0