Lim x -> 0 cot2x / cscx: can't get past cos(2x)/sin(2x)/...

[fatlord]

Lim x -> 0 cot2x / cscx

I'm getting

cos(2x)/sin(2x) / (1/sin(x))

i'm having trouble reducing after that. help me out

 

[galactus]

\(\displaystyle \lim_{x\to 0}\frac{cot(2x)}{csc(x)}\)

\(\displaystyle \frac{cot(2x)}{csc(x)}=\frac{cos(2x)}{sin(2x)}\cdot sin(x)=\frac{cos(2x)sin(x)}{2sin(x)cos(x)}=\frac{cos(2x)}{2cos(x)}\)

You can rewrite as \(\displaystyle \lim_{x\to 0}\frac{cos(2x)}{2cos(x)}\)

Now, can you see the limit?. Just plug in x=0

 

[fatlord]

1/2