Euler Formula for Sine Function/Complex Numbers/Proof 1/Lemma 1

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 $x \ge 0$.

$\blacksquare$