Euler Formula for Sine Function/Complex Numbers/Proof 1/Lemma 1
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Theorem
The function:
- $\dfrac {\sinh x} x$
is increasing for positive real $x$.
Proof
Let $\map f x = \dfrac {\sinh x} x$.
By Quotient Rule for Derivatives and Derivative of Hyperbolic Sine:
- $\map {f'} x = \dfrac {x \cosh x - \sinh x} {x^2}$
From Hyperbolic Tangent Less than X, we have $\tanh x \le x$ for $x \ge 0$.
Since $\cosh x \ge 0$, we can rearrange to get $x \cosh x - \sinh x \ge 0$.
Since $x^2 \ge 0$, we have $\map {f'} x \ge 0$ for $x \ge 0$.
So by Derivative of Monotone Function it follows that $\map f x$ is increasing for $x \ge 0$.
$\blacksquare$