Jonah Jacobson

2022-09-12

What are the asymptotes of $y=\frac{1}{x-2}$ and how do you graph the function?

Lorenzo Aguilar

Beginner2022-09-13Added 18 answers

$y=\frac{1}{x-2}$

y is defined for $x\in (-\infty ,2)\cup (2,+\infty )$

Consider $\underset{x\to {2}^{+}}{lim}y=+\infty$

And $\underset{x\to {2}^{-}}{lim}y=-\infty$

Hence, y has a vertical asymptote x=2

Now, consider $\underset{x\to \infty}{lim}y=0$

Hence, y has a horizontal asymptote y=0

y is a rectangular hyperbola with the graph below.

graph{1/(x-2) [-10, 10, -5, 5]}

y is defined for $x\in (-\infty ,2)\cup (2,+\infty )$

Consider $\underset{x\to {2}^{+}}{lim}y=+\infty$

And $\underset{x\to {2}^{-}}{lim}y=-\infty$

Hence, y has a vertical asymptote x=2

Now, consider $\underset{x\to \infty}{lim}y=0$

Hence, y has a horizontal asymptote y=0

y is a rectangular hyperbola with the graph below.

graph{1/(x-2) [-10, 10, -5, 5]}

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