📂Lemmas

The Limit of 1-cos(x)/x

Formula

$$ \lim \limits_{x \to 0} \dfrac{1 - \cos x}{x} = 0 $$

Proof

$$ \begin{align*} \lim \limits_{x \to 0} \dfrac{1 - \cos x}{x} &= \lim \limits_{x \to 0} \dfrac{1 - \cos x}{x} \dfrac{1 + \cos x}{1 + \cos x} \\ &= \lim \limits_{x \to 0} \dfrac{1 - \cos^{2} x}{x(1+\cos x)} \\ &= \lim \limits_{x \to 0} \dfrac{\sin^{2}x}{x(1+\cos x)} \\ &= \lim \limits_{x \to 0} \dfrac{\sin x}{x} \dfrac{\sin x}{1+\cos x} \\ &= \lim \limits_{x \to 0} \dfrac{\sin x}{x} \cdot \lim \limits_{x \to 0} \dfrac{\sin x}{1+\cos x} \\ &= 1 \cdot \dfrac{0}{2} \\ &= 0 \\ \end{align*} $$

Limit of the Sine Function

$$ \lim \limits_{x \to 0} \dfrac{\sin x}{x} = 1 $$

${}$

■