# How do you graph (6x)/(x^2-36) using asymptotes, intercepts, end behavior?

May 29, 2017

$\textcolor{b r o w n}{\text{Very detailed explanation given.}}$
Vertical asymptotes at $x = \pm 6$
As $x \to - \infty : y \to {0}^{-}$
As $x \to + \infty : y \to {0}^{+}$

#### Explanation:

Let a minute amount of $x$ be designated by the symbol $\delta x$
I have deliberately used this symbol as in introduction to calculus without actually using calculus

Set to $\text{ } y = \frac{6 x}{{x}^{2} - 36}$

Notice that the denominator is the difference of two squares. So we have:

$y = \frac{6 x}{\left(x - 6\right) \left(x + 6\right)}$

Note that the use of $\to$ in this explanation is used as meaning 'tends to'.

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$\textcolor{b l u e}{\text{Determine the vertical asymptotes}}$

This becomes 'undefined if the denominator is zero thus we have vertical asymptotes at those points.

Vertical asymptotes at $x = \pm 6$
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$\textcolor{b l u e}{\text{Determine the behaviour either side of the asymptotes}}$

$\textcolor{b r o w n}{\text{Set } x = + 6}$

Let $\delta x > 0$ but minute

Set $y = \frac{6 \left(x + \delta x\right)}{\left(\textcolor{w h i t e}{.} \left[x + \delta x\right] - 6\right) \textcolor{w h i t e}{.} \left(\textcolor{w h i t e}{.} \left[x + \delta x\right] + 6\right)}$

Now $\delta x$ is so small that it is almost not there so
$\left(\textcolor{w h i t e}{.} \left[6 + \delta 6\right] - 6\right)$ is positive but so small then it almost zero. Consequently the whole denominator is only just bigger than 0 but on the positive side. Dividing this into $6 \left(6 + \delta 6\right)$ yields a number that is tending to $+ \infty$

...................................................................................................
Let $\delta x < 0$ but minute

Following the same methodology of thinking we find that in this case the denominator is only just smaller than 0 so when divide into the numerator the whole is tending to $- \infty$
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$\textcolor{b r o w n}{\text{Set } x = - 6}$

As in the processes above:

When $\delta x > 0$ we have $\frac{6 \left(6 + \delta 6\right)}{{0}^{+}} \to + \infty$
When $\delta x < 0$ we have $\frac{6 \left(6 + \delta 6\right)}{{0}^{+}} \to - \infty$
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$\textcolor{b l u e}{\text{Determine the behaviour at the extremities } x \to \pm \infty}$

As $x$ becomes increasingly larger and larger the -36 has less and less effect. So the equation behaves as if it where $y = 6 \frac{x}{x} ^ 2 = \frac{6}{x}$

The net consequence is that $y \to \pm \infty$ in keeping the sign of $x$

For $x < 0 : y \to - \infty$
For $x > 0 : y \to + \infty$