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Evaluate the limit as x goes to 0 of (1 – cos (x2)) / (x2 sin (x2))

by RoRi

Evaluate the limit.

    \[ \lim_{x \to 0} \frac{1 - \cos (x^2)}{x^2 \sin(x^2)}. \]

We use the expansions (p. 287) for \cos x and \sin x as x \to 0 to write,

    \begin{align*} \cos (x^2) &= 1 - \frac{x^4}{2!} + o(x^5) \\ \sin (x^2) &= x^2 + o(x^4) \\ \end{align*}

Therefore, we compute the limit as

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

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