Limit of Sine of X over X at Zero/Proof 1

Theorem

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

Proof

\(\ds \sin x\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\)

Definition of Real Sine Function

\(\ds \)

\(=\)

\(\ds \left({-1}\right)^0 \frac{x^{2 \cdot 0 + 1} } { \left({2 \cdot 0 + 1}\right)!} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\)

\(\ds \)

\(=\)

\(\ds x + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\)

\(\ds \lim_{x \mathop \to 0} \frac {\sin x} x\)

\(=\)

\(\ds \lim_{x \mathop \to 0} \frac {x + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} } x\)

\(\ds \)

\(=\)

\(\ds \lim_{x \mathop \to 0} \frac x x + \lim_{x \mathop \to 0} \frac{\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} } x\)

\(\ds \)

\(=\)

\(\ds 1 + \lim_{x \mathop \to 0} \frac {\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} } 1\)

Power Series is Differentiable on Interval of Convergence and L'Hôpital's Rule

\(\ds \)

\(=\)

\(\ds 1 + \lim_{x \mathop \to 0} \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}\)

\(\ds \)

\(=\)

\(\ds 1 + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {0^{2 n} } {\paren {2 n}!}\)

Real Polynomial Function is Continuous

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$