How do you find the limit of $(x+1-cos(x))/(4x)$ as x approaches 0?

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Answer 1 · 2016-05-30T12:32:53

$lim_{x->0}(x + 1 - Cos(x))/(4 x) = 1/4$

We will be using

$Cos(a+b)=Cos(a)Cos(b) - Sin(a) Sin(b)$

and

$sin^2(a) + cos^2(a) = 1$

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

and substituting

$(x + 1 - Cos(x))/(4 x) = (x +2sin^2 (x/2))/(4x)$
$lim_{x->0}(x +2sin^2 (x/2))/(4x)=lim_{x->0}(x/(4x))+lim_{x->0}(2 sin^2 (x/2))/(4x)$

but

$lim_{x->0}(2 sin^2 (x/2))/(4x) = lim_{x->0}sin(x/2)/4lim_{x->0}(sin(x/2)/(x/2)) = 0 times 1$

so

$lim_{x->0}(x + 1 - Cos(x))/(4 x) = 1/4$

How do you find the limit of (x+1-cos(x))/(4x)(x+1-cos(x))/(4x) as x approaches 0?