find the lim (cos x-1)/ (sin 2x)

Question

x->0

Answer 1

can you use L'Hopital's rule?

if so
lim (cos x-1)/ (sin 2x)
x->0
= lim (-sinx)/(2cos 2x) as x ---> 0
= 0/2 = 0

Answer 2

no u cant

Answer 3

some common trig limits you should have come across:
lim( (sinx)/x ) as x --> 0 = 1
lim( x/sinx ) as x --> 0 = 1
lim( (cosx - 1)/x) as x -- 0 = 0

so lets rewrite the question as

lim[ ((cos x-1)/x) * x(2sinxcosx) ]
= lim( (cosx - 1)/x) * lim (x/sinx) * lim (1/2cosx)
= 0(1)(1/2) = 0