# For what values of x, if any, does f(x) = 1/((x-5)sin(pi+1/x)  have vertical asymptotes?

Feb 24, 2018

$\frac{1}{\left(x - 5\right) \sin \left(\pi + \frac{1}{x}\right)}$ has vertical asymptotes at $x = 5$ and $x = \frac{1}{m \pi}$

#### Explanation:

Vertical asymptotes of $\frac{1}{\left(x - 5\right) \sin \left(\pi + \frac{1}{x}\right)}$ are there when

either $x - 5 = 0$ i.e. $x = 5$

or when $\sin \left(\pi + \frac{1}{x}\right) = 0$ i.e. $\pi + \frac{1}{x} = n \pi$, where $n$ is an integer

or $\frac{1}{x} = \pi \left(n - 1\right)$ or $x = \frac{1}{\pi \left(n - 1\right)} = \frac{1}{m \pi}$, where $m$ is an integer so that $m = n - 1$. Note that as $m$ increases, value of $x$ decreases continuously and maximum value (less than $5$) is $x = \frac{1}{\pi} = 0.3183$

graph{1/((x-5)sin(pi+1/x)) [-0.996, 0.996, -0.498, 0.498]}

graph{1/((x-5)sin(pi+1/x)) [-14.41, 17.46, -8.86, 7.08]}