SOLUTION: Why do rational expressions produce asymptote?

Question 246331: Why do rational expressions produce asymptote?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
I assume you mean: "Why do rational functions produce vertical asymptotes?" If not, stop reading and repost with a clearer question.

Before I explain why, I want to make the point that not all rational functions have vertical asymptotes. Many do but not all.

Rational functions that have vertical asymptotes are ones that have denominators which could be zero for certain values of x. There are vertical asymptotes for these functions for each value of x that would make any denominator zero.

Remember that vertical lines have equations like x = 5 and they represent all the points with a certain x coordinate.

So why do we get vertical asymptotes for x values that make a denominator zero? Let's consider the two main characteristics of vertical asymptotes:

The graph of a function does not cross (or intersect with) a vertical asymptote. To understand the reason this is true, let's consider the possibility that a function does cross a vertical asymptote. If it did, then the x coordinate of that point would be one of the x values that makes a denominator of the function zero. And zero denominators are one of the things we can never allow to happen! This is why functions do not cross their vertical asymptotes.
For x values very close to the ones that make a denominator zero, the function has very high positive or negative values. In other words, the points on the graph that are very near a vertical asymptote are either very high or very low. The reason for this is based on how fractions work. Larger denominators make the value of a fraction smaller. And denominators that are close to zero make the value of a fraction very large. For x values that are very close to the ones that make the denominator zero, the denominators will be very close to zero which makes the value of the whole fraction very large (positive or negative). (If this is not clear, just try a few fractions on your calculator with denominators that are close to zero. For example, see how 1/0.001, 1/0.0001 and 1/0.00001 keep getting progressively larger and larger as the denominators get closer and closer to zero.)

Here's a sample function and its graph so you can see some of this "in action".

The values of x that make this denominator zero are 1 and 4, Below is the graph. (Normally vertical asymptotes are drawn as dotted lines. But Algebra.com's graphing software does not draw the vertical asymptotes so you will just have to imagine the asymptotes which are the vertical lines x = 1 and x = 4.)

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