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### Course: Get ready for AP® Calculus>Unit 1

Lesson 21: Discontinuities of rational functions

# Discontinuities of rational functions

Sal analyzes two rational functions to find their vertical asymptotes & removable discontinuities. He distinguishes those from the zeros of the functions.

## Want to join the conversation?

• Why does a value that makes the denominator equal to 0 count as a vertical asymptote? Shouldn't it be a removable discontinuity? And also why is it that the removable discontinuity happens to be a factor shared in both the denominator and the numerator?
• When a value makes the denominator of a rational function equal to zero, it results in an undefined value for the function. This situation often leads to a vertical asymptote rather than a removable discontinuity. Let me explain why.

Vertical Asymptote:

When the denominator of a rational function becomes zero, it indicates that the function is undefined at that point.
A vertical asymptote is a vertical line that the graph approaches but never crosses.
If a function has a vertical asymptote at a certain x-value, it means the function becomes unbounded (either positive or negative) as it approaches that x-value from one side or the other.
Removable Discontinuity:

A removable discontinuity, also known as a hole, occurs when a factor is common to both the numerator and denominator of a rational function, causing a cancellation.
If a factor

(x−a) is in both the numerator and denominator, it can be canceled out, resulting in a simplified function without the discontinuity.
Now, regarding your question about why the removable discontinuity often involves a common factor in the numerator and denominator:

This happens because when you factor a polynomial, any common factor in both the numerator and denominator can be canceled out without affecting the value of the function at points where the denominator is not zero.
When the common factor is canceled, it results in a simplified function that is defined at the point where the factor was previously causing a discontinuity.
In summary, a value that makes the denominator zero typically leads to a vertical asymptote, but if that zero is also a factor in the numerator, it can be canceled out, resulting in a removable discontinuity. The distinction depends on whether the cancellation occurs or not.
• what is a asymptote and what is the difference between asymptote and removable discountinuity ? please help and thanks in advance !
• One definition of an asymptote of a curve is that it is a LINE such that the distance between the curve and the line approaches zero as they tend to infinity.

Or, in simple terms, you could think of an asymptote as a LINE that a curve approaches but never meets. (Later on in your math career, you will discover that vertical asymptotes are as I've described, but that sometimes the function does actually cross through its horizontal asymptote(s).)

A removable discontinuity is a SINGLE POINT for which the function is not defined. If you were graphing the function, you would have to put an open circle around that point to indicate that the function was not defined there.
Hope this is of some help!
• x=4 should it be removable discontinuity?
• The vertical asymptote(s) can only be found once the equation is as simplified as possible. Removable discontinuities are found as part of the simplification process. If a factor like x=4 appears in both steps the vertical 'asymptote' label is the stronger since it produces a vertical asymptote when graphed as Sal shows.
• Can a point have both removable discontinuity and vertical asymptote?
• No. A vertical asymptote is when a rational function has a variable or factor that can be zero in the denominator.
A hole is when it shares that factor and zero with the numerator.
So a denominator can either share that factor or not, but not both at the same time. Thus defining and limiting a hole or a vertical asymptote.
• Just to Clarify, Will a removable discontinuity always contain an extraneous solution?
• A removable discontinuity always contain an extraneous solution.
• Sorry, this sounds like a dumb question but I'll ask anyway. If you have two expressions like in the video above, and they cancel each other out, then the number that makes those two equal to zero is the removable discontinuity.
• Yes, exactly. You can see the top answer for details:)
• How come he doesn't use the abc-formula to use a more certain way to find the numbers when simplifying the function?
• Because he just wanted to use another way of working out the values for x. Also, if he didn't factor, then he wouldn't get the part where he states that x=-4 is a removable discontinuity, but x=-6 is not.
• What is a (Removable Discontinuity)?
• Also called a hole, it is a spot on a graph that looks like it is unbroken that actually has nothing there, a hole in the line. the simplest example is x/x. if you graphed it it would look like y=1, but if you tried to plug in 0 you would get undefined, so there is a hole at x=0, or a removable discontinuity.

Let me know if that doesn't make sense.
• couldn't it be argued that if you plug in 6 for x (second example in the video) it too would be undefined, and therefore a removable discontinuity?
• No. The function isn't undefined when x=6
(6-6)/(6+6) = 0/12 = 0
Hope this helps.
• how do we know if its a vertical or horizantal asymtope.
(1 vote)
• To determine whether a function has a vertical or horizontal asymptote, we need to analyze its behavior as x approaches infinity or negative infinity. Here are the general steps to determine the type of asymptote:

1. Determine the degree of the numerator and denominator of the rational function.

2. If the degree of the numerator is less than the degree of the denominator, then the x-axis (y=0) is the horizontal asymptote.

3. If the degree of the numerator is greater than or equal to the degree of the denominator, then perform polynomial long division to reduce the function to the form of a polynomial plus a proper rational function.

4. If the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator after performing long division, then the x-axis (y=0) is the horizontal asymptote.

5. If the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator after performing long division, then the line y = a/b (where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator) is the horizontal asymptote.

6. If the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator after performing long division, then there is no horizontal asymptote.

7. To find vertical asymptotes, we need to find the values of x that make the denominator equal to zero, but not the numerator.

8. If the denominator factors into linear factors, then each factor corresponds to a vertical asymptote.

9. If the denominator has an irreducible quadratic factor, then we have a vertical asymptote if and only if the numerator has a degree less than the degree of the denominator.

10. If the denominator has a repeated linear factor, then we have a vertical asymptote if and only if the numerator has a degree less than one more than the degree of the denominator.

In summary, the type of asymptote (vertical or horizontal) is determined by the degree of the numerator and denominator of the function, and their behavior as x approaches infinity or negative infinity.