# How do you find the asymptotes for f(x)=sinx/(x(x^2-81))?

Mar 12, 2016

Horizontal asymptote:

$y = 0$

Vertical asymptotes:

$x = - 9$

$x = 9$

#### Explanation:

When we factorize the denominator, we can write

f(x) = frac{sin(x)}{(x+9)x(x-9)

For this kind of function, we have to check for the points where the denominator is zero, as there cannot be division by zero. Also we need to check for $\pm \infty$.

Since the numerator fluctuates about -1 to 1, while the denominator keeps increasing in magnitude, we know that

${\lim}_{x \to \infty} f \left(x\right) = 0$
${\lim}_{x \to - \infty} f \left(x\right) = 0$

There is a horizontal asymptote: $y = 0$

The denominator equals zero when $x = - 9$, $x = 0$ or $x = 9$. We check them one by one.

First, we check the behavior of $f \left(x\right)$ when $x$ is in the region of $- 9$. We know that $\sin \left(9\right) \approx 0.412 > 0$.

${\lim}_{x \to - {9}^{-}} \frac{1}{x \left({x}^{2} - 81\right)} = \infty$
${\lim}_{x \to - {9}^{+}} \frac{1}{x \left({x}^{2} - 81\right)} = - \infty$

Therefore,

${\lim}_{x \to - {9}^{-}} f \left(x\right) = \infty$
${\lim}_{x \to - {9}^{+}} f \left(x\right) = - \infty$

There is a vertical asymptote: $x = - 9$

Next, we check the behavior of $f \left(x\right)$ when $x$ is in the region of $0$.

Since $f \left(x\right)$ is of the indeterminate form of $\frac{0}{0}$, we apply the L'hospital Rule.

${\lim}_{x \to 0} f \left(x\right) = {\lim}_{x \to 0} \frac{\sin \left(x\right)}{{x}^{3} - 81 x}$

= lim_{x->0} frac{frac{"d"}{"d"x}(sin(x))}{frac{"d"}{"d"x}(x^3-81x)}

$= {\lim}_{x \to 0} \frac{\cos \left(x\right)}{3 {x}^{2} - 81}$

$= \frac{\cos \left(0\right)}{3 {\left(0\right)}^{2} - 81}$

$= - \frac{1}{81}$

Seems like $f \left(x\right)$ is continuous at $x = 0$ and there is no asymptote.

Now, rather than going through the same process and check for $x = 9$, I'm going to say that $f \left(x\right)$ is an even function. That means

$f \left(x\right) = f \left(- x\right)$

for every $x$ in the domain of $f \left(x\right)$. (Try proving this yourself) Therefore, the graph of $y = f \left(x\right)$ will be symmetrical about the $y$-axis.

Since there is a vertical asymptote of $x = - 9$, the other side is going to have its "reflection". The reflection of $x = - 9$ about $x = 0$ (the $y$-axis) is $x = 9$. Hence, there is a vertical asymptote: $x = 9$

Here is a graph of $y = f \left(x\right)$ for your reference.
graph{sin(x)/(x^3-81x) [-20, 20, -0.08, 0.08]}