Question

evaluate limit -> 0 sin^2(pix) / 3x^2? the pi throws me off

Answer 1

as x goes to zero?\[ \lim_{x \rightarrow 0}[\sin^2 (\pi x)/3x^2]\]
like so?

Answer 2

yess

Answer 3

take derivative from nominator and denominator
2sin(pix)*pi*cos(pix) / 6x
again :
(pi/3) * ((picos^2(pix) - pisin^2(pix)/1)

(pi^2 / 3)

Answer 4

use L'Hospital's Rule I think
derivative of top and bottom:
\[2 \pi \sin (\pi x)\cos (\pi x)\]
(top)
6x on the bottom

Answer 5

Coolsector is right. I got the same

Answer 6

you can as well :

based on the limit lim x->0 sin(kx)/kx = 1

multiply by pi^2/pi^2 to get :

pi^2sin^2(pix) / 3pi^2x^2

now : pi^2/3 * lim(sin(pix)/pix)) * (lim(sin(pix)/pix) = pi^2/3