Limit of Sine of X over X/Proof 1

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Theorem

$\displaystyle \lim_{x \mathop \to 0} \frac {\sin x} x = 1$


Proof

\(\displaystyle \sin x\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\) Definition of Real Sine Function
\(\displaystyle \) \(=\) \(\displaystyle \left({-1}\right)^0 \frac{x^{2 \cdot 0 + 1} } { \left({2 \cdot 0 + 1}\right)!} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\)
\(\displaystyle \) \(=\) \(\displaystyle x + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\)


\(\displaystyle \lim_{x \mathop \to 0} \frac {\sin x} x\) \(=\) \(\displaystyle \lim_{x \mathop \to 0} \frac {x + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} } x\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{x \mathop \to 0} \frac x x + \lim_{x \mathop \to 0} \frac{\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} } x\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + \lim_{x \mathop \to 0} \frac {\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} } 1\) Power Series is Differentiable on Interval of Convergence and L'Hôpital's Rule
\(\displaystyle \) \(=\) \(\displaystyle 1 + \lim_{x \mathop \to 0} \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {0^{2 n} } {\paren {2 n}!}\) Real Polynomial Function is Continuous
\(\displaystyle \) \(=\) \(\displaystyle 1\)

$\blacksquare$