Find the limit as x goes to 0 of (cos (sin x) - cos x) / x4

Evaluate the limit.

$\lim_{x \to 0} \frac{\cos (\sin x) - \sin x}{x^4}.$

We know (page 287 of Apostol) that the expansions for $\sin x$ and $\cos x$ as $x \to 0$ are given by

$\begin{align*} \sin x &= x - \frac{x^3}{6} + o(x^4) \\ \cos x &= 1 - \frac{x^2}{2} + \frac{x^4}{24} + o(x^5). \end{align*}$

Therefore, we have the following expansion for $\cos (\sin x)$ as $x \to 0$ ,

$\begin{align*} \cos (\sin x) &= 1 - \frac{(\sin x)^2}{2} + \frac{(\sin x)^4}{24} + o((\sin x)^5) \\[9pt] &= 1 - \frac{1}{2} \left( x - \frac{x^3}{6} + o(x^4) \right)^2 + \frac{1}{24} \left( x - \frac{x^3}{6} + o(x^4) \right)^4 + o(x^5) \\[9pt] &= 1 - \frac{1}{2} \left( x^2 - \frac{x^4}{3} + o(x^4) \right) + \frac{1}{24} \left(x^4 + o(x^4)\right) + o(x^5) \\[9pt] &= 1 - \frac{1}{2}x^2 + \frac{5}{24}x^4 + o(x^4). \end{align*}$

So, now we can take the limit,

$\begin{align*} \lim_{x \to 0} \frac{\cos (\sin x) - \sin x}{x^4} &= \lim_{x \to 0} \frac{1 - \frac{1}{2}x^2 + \frac{5}{24}x^4 + o(x^4) - 1 + \frac{1}{2}x^2 - \frac{1}{24}x^4 + o(x^5)}{x^4} \\[9pt] &= \lim_{x \to 0} \frac{\frac{1}{6} x^4 + o(x^4)}{x^4} \\[9pt] &= \lim_{x \to 0} \left( \frac{1}{6} + \frac{o(x^4)}{x^4} \right) \\[9pt] &= \frac{1}{6}. \end{align*}$

Stumbling Robot

Find the limit as x goes to 0 of (cos (sin x) – cos x) / x⁴

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