Limit of Tan X over X at Zero/Proof 2
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Theorem
- $\ds \lim_{x \mathop \to 0} \frac {\tan x} x = 1$
Proof
| \(\ds \lim_{x \mathop \to 0} \frac {\tan x} x\) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac 1 {\cos x} \frac {\sin x} x\) | Definition of Tangent Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac 1 {\cos x} \lim_{x \mathop \to 0} \frac {\sin x} x\) | Product Rule for Limits of Real Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac {\sin x} x\) | Cosine of Zero is One | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) | Limit of $\dfrac {\sin x} x$ at Zero |
$\blacksquare$