13 Limiting Ratios.

Article 13
Limiting Ratios.

To ﬁnd the limit, as $u$ approaches zero, of

\frac{sinh u}{u}, \frac{tanh u}{u},

which are then indeterminate in form.

By eq. (14), $sinh u > u > tanh u$ ; and if $sinh u$ and $tanh u$ be successively divided by each term of these inequalities, it follows that

\begin{matrix} 1 < \frac{sinh u}{u} < cosh u, \\ sech u < \frac{tanh u}{u} < 1, \end{matrix}

but when $u ≐ 0$ , $cosh u ≐ 1$ , $sech u ≐ 1$ , hence

lim_{u ≐ 0} \frac{sinh u}{u} = 1, lim_{u ≐ 0} \frac{tanh u}{u} = 1 .

(24)