📂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

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

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