# What are all horizontal asymptotes of the graph y=(5+2^x)/(1-2^x) ?

Sep 22, 2014

Let us find limits at infinity.

${\lim}_{x \to + \infty} \frac{5 + {2}^{x}}{1 - {2}^{x}}$

by dividing the numerator and the denominator by ${2}^{x}$,

$= {\lim}_{x \to + \infty} \frac{\frac{5}{2} ^ x + 1}{\frac{1}{2} ^ x - 1} = \frac{0 + 1}{0 - 1} = - 1$

and

${\lim}_{x \to - \infty} \frac{5 + {2}^{x}}{1 - {2}^{x}} = \frac{5 + 0}{1 - 0} = 5$

Hence, its horizontal asymptotes are

$y = - 1$ and $y = 5$

They look like this: