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

Theorem
The function:


 * $\dfrac {\sinh x} x$

is increasing for $x \ge 0$.

Proof
Let $f \left({x}\right) = \dfrac {\sinh x} x$.

By Quotient Rule for Derivatives and Derivative of Hyperbolic Sine Function:


 * $f' \left({x}\right) = \dfrac {x \cosh x - \sinh x} {x^2}$

By Hyperbolic Tangent Less than X, we have $f \left({x}\right) \ge 0$.

So by Derivative of Monotone Function it follows that $f \left({x}\right)$ is increasing.