A limit via the cosine series

MediumCalculus~3m

Evaluate $\displaystyle\lim_{x\to 0}\frac{\cos x - 1 + \tfrac{x^2}{2}}{x^4}$ . Give a decimal to four places.

Show hints (2)+

One-term $\cos x\approx 1-\tfrac{x^2}{2}$ is too crude (gives $0$ ) - expand to the $x^4$ term.
$\cos x - 1 + \tfrac{x^2}{2} = \tfrac{x^4}{24}+\cdots$ ; the limit is the coefficient $\tfrac1{4!}$ .

Answer

Reveal answer →

0.0417 (± 0.0005)

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