Find the domain, vertical asymptotes, and horizontal asymptotes of the function. f(x)=x/x2−81

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 ±∞ .

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

lim

x->∞ f(x)=0

lim

x-> - ∞ f(x)=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(x)when x is in the region of −9. We know that sin(9)≈0.412>0.

There is a vertical asymptote:

x=−9 Next, we check the behavior of f(x)when x is in the region of 0.

Since f(x) is of the indeterminate form of 0,0 we apply the L'hospital Rule.

Seems like f(x) 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(x) is an even function. That means f(x)=f(−x).

for every x in the domain of f(x). (Try proving this yourself) Therefore, the graph of y=f(x) 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(x) for your reference.

graph{sin(x)/(x^3-81x) [-20, 20, -0.08, 0.08]}