Vertical & Horizontal Aymptotes

Question

Find all vertical & horizontal asymptotes of the graph of the function :

a. f(x) = 4/x²

b. f(x) = x³/x² - 1

Answer 1

y = 4/x²

For this one, start by assuming that x = 0. The value of y becomes undefined (approaching infinity). For any other value of x, there is a defined value of y. For negative x values, y also goes towards positive infinity.

As x becomes large, y tends to 0

So, 2 asymptotes: x = 0, and y = 0

"Asymptote" means a value which is approached, but never actually reached. In this case, you can see it by just plugging in decreasing numbers for x.

To visualize things like this, it is very useful to use a graphing calculator or online graphing tool.

y = x3/x2 - 1

Did you mean this? y = x³/(x² - 1)

Otherwise, it would simplify to x-1, which has no asymptotes.

Let's assume you meant:

y = x³/(x² - 1)

If x is 1, the bottom goes to 0, and y goes to infinity. The same thing happens when x goes to -1.

There is also an asymptote as x becomes large---for large x, the -1 becomes less significant, and the function approaches y = s

So, 3 asymptotes: x = 1, x = -1, and y = x

Again, make the graph to visualize what is happening

Answer 2

The best way to "see" the answer to this question is to graph the 2 functions involved.

4/x² goes to 0 as x goes to + or minus infinity and it blows up (goes to +infinity) at x=0.

x³/(x²-1) blows up (in both directions) at +1 and -1 and becomes asymptotic to y=x as x goes to +infinity or -infinity.