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Find the limit as x goes to 0 of (1 – cos2 x) / (x tan x)

by RoRi

Evaluate the limit.

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

To evaluate this we use the trig identity 1 - \cos^2 x = \sin^2 x to simplify

    \begin{align*} \lim_{x \to 0} \frac{1 - \cos^2 x}{x \tan x} &= \lim_{x \to 0} \frac{\sin^2 x}{x \tan x} \\[9pt] &= \lim_{x \to 0} \frac{\sin^2 x \cos x}{x \sin x} \\[9pt] &= \lim_{x \to 0} \left( \frac{\sin x}{x} \cdot \cos x \right) \\[9pt] &= 1 \cdot 1 = 1. \end{align*}

We have proved the limit \lim_{x \to 0} \frac{\sin x}{x} = 1 earlier (at least once), but let’s do it again using the techniques of this section and o-notation.

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

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