How do you find the limit of $(1-cosx) / x$ as x approaches 0?

Question

156428 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-03T15:41:00

$0$

$1-cosx=2sin^2(x/2)$ so

$(1-cos x)/x=(x/4) (sin(x/2)/(x/2))^2$ then

$lim_(x->0)(1-cos x)/x equiv lim_(x->0)(x/4) (sin(x/2)/(x/2))^2 = 0 cdot 1 = 0$

Answer 2 · 2016-10-03T16:25:09

Multiply by $(1+cosx)/(1+cosx)$ to get

$(1-cos^2x)/(x(1+cosx)) = (sin^2x)/(x(1+cosx))$

$= sinx * sinx/x * 1/(1+cosx)$

Taking the limit as $xrarr0$ gives

$(0)(1)(1/2) = 0$

How do you find the limit of (1-cosx) / x(1-cosx) / x as x approaches 0?