# How do you find the asymptote and graph y=1/(x−5)−4?

How do you find the asymptote and graph $y=\frac{1}{x-5}-4$?
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Peyton Atkins
$y=\frac{1}{x-5}-4$
Notice y is undefined at x=5
Consider, $\underset{x\to {5}^{-}}{lim}y=-\infty \phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}\underset{x\to {5}^{+}}{lim}y=+\infty$
Also, $\underset{x\to -\infty }{lim}y=-4\phantom{\rule{1ex}{0ex}}\text{and}\phantom{\rule{1ex}{0ex}}\underset{x\to +\infty }{lim}y=-4$
Hence y is a rectangular hyperbola with a vertical asymptote of x=5 and a horizontal asymptote of y=−4
The graph of y is shown below.
graph{1/(x-5)-4 [-5.42, 17.08, -7.105, 4.14]}