# What are the asymptotes for  f(x)=x/(x-7)?

Sep 21, 2015

The vertical assymptote is $x = 7$ and the horizontal assymptote is $y = 1$

#### Explanation:

Vertical assymptotes are the lines of the type $x = a$ that the function can never take because doing so will create a math error.

In this function the only possible math error is a division by zero, so we have that:
$x - 7 \ne 0$
$x \ne 7$

So $7$ is a vertical assymptote. That means that as the function gets closer and closer to 7, the values of the function as whole will become bigger in magnitude (because $x - 7$ gets closer and closer to 0) but the function will never actually evaluate at that point.

Horizontal assymptotes are the lines of the type $y = b$ that are basically the value the function start to take whenever $x$ becomes bigger and bigger. Slant assymptotes (of the type $y = a x + b$) are very similar but the function isn't tending to a constant value. We can discover them the same way:

Start plugging bigger and bigger values until you see a pattern, or, use a made-up number, like, let's say $b$, that is infinitely big, plug it in and see what happens.

$y = \frac{b}{b - 7}$

Since $b$ is an infinitely large number, we can say that $b - 7 \cong b$, after all think of it like if b was a number like a billion or a trillion. A trillion minus 7 is still pretty much a trillion for practical purposes. So we have

$y \cong \frac{b}{b}$ or $y \cong 1$

So the horizontal assymptote is $y = 1$, that is, as $x$ grows bigger the function gets closer and closer to $1$.

We can doublecheck it by looking at the graph:
graph{x/(x-7) [-14, 14, -28, 28]}