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Find the limit as x goes to 0 of (sin x) / (arctan x)

Evaluate the limit.

    \[ \lim_{x \to 0} \frac{\sin x}{\arctan x}. \]


We know (p. 287) the following expansions as x \to 0,

    \[ \sin x = x + o(x^2), \qquad \text{and} \qquad \arctan x = x + o(x^2). \]

(Note that these are the same expansion when we use only the first order terms. This tells us that \sin x and \arctan x behave similarly near 0. We would need to take higher order terms to differentiate between the two. For instance, if we wanted to include cubic terms we would have \sin x = x - \frac{1}{6}x^3 + o(x^4), but \arctan x = x - \frac{1}{3}x^3 + o(x^4).) From here we compute the limit,

    \[ \lim_{x \to 0} \frac{\sin x}{\arctan x} = \lim_{x \to 0} \frac{x + o(x^2)}{x+o(x^2)} = 1. \]