find the limit?

Question

lim (sin2h) (sin3h)/(x^2)
x->0

Answer 1

Assuming you meant sin(2x)sin(3x)/x^2 (not h).

We know that sin(x)/x -> 1 as x->0.

So sin(2x)/2x -> 1 as x->0. So:

sin(2x)/x -> 2 as x->0.

Similarly:

sin(3x)/3x -> 1 as x->0

so:

sin(3x)/x -> 3 as x->0.

So sin(2x)sin(3x)/x^2 = sin(2x)/x * sin(3x)/x -> 2*3 = 6.

Answer 2

Unless h is a function of x, this blows up and becomes infinite as x --> 0

Answer 3

lim (sin2x) (sin3x)/(x^2)
x->0

=lim x-->0 (sin2x)/x lim x-->0 (sin3x)/x
=(2 lim 2x-->0 (sin2x)/2x) (3 lim 3x-->0 (sin3x)/3x)
= (2)(3) = 6