Evaluating \(\displaystyle\lim_{{{x}\to{0}}}{\frac{{{1}-{\cos{{x}}}}}{{{\left({2}{x}\right)}^{{2}}}}}\) I have this question in my

Question

Evaluating

lim_{x \to 0} \frac{1 - \cos x}{{(2 x)}^{2}}

I have this question in my book and the answer amounts to

\frac{1}{4}

. I don't see how, and I think it's an error. Can you confirm it is an error?

lim_{x \to 0} \frac{1 - \cos x}{{(2 x)}^{2}}

I assume it would need to be

\cos^{2} x

instead.

Answer 1

It's indeed not correct. Simply use l'Hopital rule allow you to conclude that

lim_{x \to 0} \frac{1 - \cos (x)}{{(2 x)}^{2}} = lim_{x \to 0} \frac{1 - \cos (x)}{4 x^{2}} = lim_{x \to 0} \frac{\sin (x)}{8 x} = \frac{1}{8}

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