can you solve this lim [ 1-cos((cos(2x))^(1/2))]/[x^2] when x--->0?

Question

can you solve this lim [ 1-cos((cos(2x))^(1/2))]/[x^2] when x--->0

Answer 1

This isn't too hard if you've written it correctly.
cos(2x) goes to 1 smoothly, so cos(sqrt(cos2x)) goes to cos(1) which is about .54
1-.54 is about .45 so [1-.54]/[x^2] is discontinuous at 0, and there is no limit. The limit as x approaches infinity from either side is +infinity

Answer 2

i assume it truly is a quadratic equation, x^2 - x - a million = 0. Others have given the perfect answer, (a million +/- sqrt(5))/2. merely an engaging addendum to this: the useful root right here, (a million+sqrt(5))/2, or about a million.618, is in truth phi, the golden ratio.