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## Integrated math 3

### Course: Integrated math 3>Unit 13

Lesson 3: 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?
• 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.
• 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.
• 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:)
• 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?