15.13. Limit of a Hyperbolic Function

A hyperbolic function are analogous to trigonometric functions but refer to base e. Below are six hyperbolic functions derived from the basic definitions of sinhx and coshx.

$\textup{sinh}\, x=\frac{e^x-e^{-x}}{2}$

$\textup{tanh}\, x=\frac{\textup{sinh}\, x}{\textup{cosh}\, x}$

$\textup{sech}\, x=\frac{1}{\textup{cosh}\, x}$

$\textup{cosh}\, x=\frac{e^x+e^{-x}}{2}$

$\textup{coth}\, x=\frac{\textup{cosh}\, x}{\textup{sinh}\, x}$

$\textup{csch}\, x=\frac{1}{\textup{sinh}\, x}$

Calculating the limit of a hyperbolic function involves evaluating a function in terms of e^x and e^(-x) as x approaches c. Like other functions, it can be found by substituting x = c, but it is similarly possible that the limit may not exist.

Illustrative Examples
Calculate each limit if it exists.

$1)\lim_{x\rightarrow 2}\textup{cosh}\, x$