Find the limit?

Question

#lim_{x->0}(cos(8x)-1)/sin(9x)#

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Answer 1 · 2018-02-28T05:31:44

# 0#.

Recall that, #1-cos2theta=2sin^2theta#

#:. (cos8x-1)/(sin9x)=(-2sin^2 4x)/(sin9x)#.

Since, #lim_(y to 0)siny/y=1#, we have,

#"The Reqd. Lim.="lim_(x to 0)(-2sin^2 4x)/(sin9x)#,

#=lim{-2*(sin4x)/(sin9x)*sin4x}#,

#=lim-2*((sin4x)/(4x)*4x)/((sin9x)/(9x)*9x)*sin4x#,

#=lim-2*4/9*((sin4x)/(4x))/((sin9x)/(9x))*sin4x#,

#rArr "The Reqd. Lim.="-2*4/9*1/1*sin(4xx0)=0#.