What is the limit of #(sin^2(9x))/(3x^2)# as x approaches 0?

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Answer 1 · 2016-08-28T13:26:48

#=27#

the trick will be to steer this into it being in terms of well known limit #lim_(x to 0) (sin x)/x = 1#

So
#lim_(x to 0) (sin^2(9x))/(3x^2)#

#= lim_(x to 0) 1/3 (sin (9x))/(x) * (sin(9x))/(x)#

#= lim_(x to 0) (9 times 9)/3 (sin (9x))/(9x) * (sin(9x))/(9x)#

let #u = 3x# and lift the constant out

#implies 27 lim_(u to 0) (sin (u))/(u) * (sin(u))/(u)#

and as the limit of the products is the product of the limits

#= 27 lim_(u to 0) (sin (u))/(u) * lim_(u to 0) (sin(u))/(u)#

#=27#