Question

How do you evaluate lim (cos(ax)-cos(bx))/(x^2) as x approaches 0 without L'Hospital's rule?

Answer 1

usually when you see something like a-b on top, and it will be undefined on bottom, multiply by the conjugate

Answer 2

in this case by\[\frac{\cos(ax)+\cos(bx)}{\cos(ax)+\cos(bx)}\]

Answer 3

try reducing to something that you are familiar.

Answer 4

what do you get after doing the multiplication and plugging in the numbers?

Answer 5

I'm getting 0. But the answer's supposed to be (b^2 - a^2)/2.

Answer 6

ok, sorry, don't plug in yet, just do the multiplication
what do you get? (use the equation editor to make it clear)

Answer 7

\[\frac{ \cos ^{2}ax - \cos ^{2}bx }{ x ^{2}(\cos ax + \cos bx) }\]

Answer 8

now apply \[\sin^2x+\cos^2x=1\]to the top