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

Question

turingtest · Answer

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

turingtest · Answer

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

experimentx · Answer

try reducing to something that you are familiar.

turingtest · Answer

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

anonymous · Answer

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

turingtest · Answer

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

anonymous · Answer

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

turingtest · Answer

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