evaluate the trigonometric Limx->0 1-cos(x/2)/x

Question

evaluate the trigonometric limits.

Answer 1

One way to solve this is use the series for the cosine function and approximate to it.

cos(x/2)=1-(x/2)²/2!+(x/2)⁴/4! -... approximates to 1-(x/2)²/2 when x→0.

1-cos(x/2) approximates to (x/2)²/2 when x→0 so (1-cos(x/2))/x becomes x/8, and x/8→0 when x→0.

So lim as x→0 of (1-cos(x/2))/x is 0.