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

##### 1 Answer

Apr 5, 2016

#lim_(x->0) sin^2(x)/(3x^2) = 1/3#

#### Explanation:

Start with your favourite proof that

That might start with a geometric illustration that for small

#sin(x) <= x <= tan(x)#

Then divide through by

#1 <= x / sin(x) <= 1 / cos(x)#

Take reciprocals and reverse the inequality (since

#cos(x) <= sin(x)/x <= 1#

Then

So

Also

Hence

Whatever method we use to find

#lim_(x->0) sin^2(x)/(3x^2) = (lim_(x->0) sin(x)/x) * (lim_(x->0) sin(x)/x) * 1/3 = 1/3#