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## Integrated math 3

### Course: Integrated math 3>Unit 13

Lesson 3: Discontinuities of rational functions

# Analyzing vertical asymptotes of rational functions

Sal analyzes the behavior of q(x)=(x²+3x+2)/(x+3) around its vertical asymptote at x=-3.

## Want to join the conversation?

• Wait a minute, q approaches +infinity when x-values are between -3 and -2. If x -> 3^+ means "values over -3," then why do we limit our values? • is simplifying the expressions eqivalent when i search for the asymptotes and their behaviour? • I didn't quite get the answer of the first question. Shouldn't q(x) be approaching to positive infinity? I mean even Sal points that out on the numberline in blue color with the arrow pointing to the right side. What's happening... I'm confused. • He starts to explain around , as q(x) approaches the vertical asymptote of -3, the function goes down and approaches negative infinity.

Try substituting any value less than -3 for x, and you'll find the function always comes out as a negative. If we look at x = -4, for example, the numerator simplifies to (-3)(-2) = 6. The denominator simplifies to -4+3 = -1. The function as a whole then simplifies to q(x) = -6 for x = -4. You can try this for any x value smaller than -3 and you'll find the function approaches negative infinity the closer x gets to it's vertical asymptote of -3.
• I would definitely start this video by considering the interval from -3 to -2 first (approaching from positive side), because if we start considering approaching from the negative, there is a temptation to use whole numbers such as (-5) and (-4) and by substituting these numbers (instead of -3.1 and -3.01) function DOES NOT appear to approach negative infinity. There is a sign change at x=-4.4. • So why do you have to restrict the -3^+ but not -3^-? • Why do we care about the intervals -3<x<-2? What does Sal mean at when he says "I don't want any kind of weird sign changes"? • How can we differentiate between horizontal and vertical asymptotes? Sometimes I hear Sal refer to horizontal asymptotes but like in this video, it focuses on vertical.
(1 vote) • Horizontal asymptotes are when a function's y value starts to converge toward something as its x value goes toward positive or negative infinity. This is the end behavior of the function.

Vertical asymptotes are when a function's y value goes to positive or negative infinity as the x value goes toward something finite.

Let's say you have the function
a(x) = (2x+1)/(x-1).

As x → 1 from the negative direction, a(x) → -∞. As x → 1 from the positive direction, a(x) → +∞. This is your vertical asymptote, because as x approaches something finite, a(x) approaches something infinite.

As x gets bigger and bigger (you can think of this as x → ∞, I don't know if you have done end behavior at this point in the course), a(x) goes to 2. You can confirm this by plugging in really big values, for example:
(2(1000000000)+1)/((1000000000)-1) = 2.000000003
Anyway, the point is that as x approaches something infinite, a(x) approaches something finite, so this is your horizontal asymptote.

Vertical asymptote: As x → finite, y → infinite
Horizontal asymptote: As y → finite, x → infinite

Hope this clarified a bit.
• In the example in the video, the line x = -3 is the vertical asymptote because it is the denominator and as x approaches -3 from either side, the denominator gets smaller at a faster rate than the rate at which the numerator is getting bigger.

If the factor (x+3) is also in the numerator, would this mean that x = -3 is no longer the vertical asymptote? Because now the numerator is changing at a pace that is just as fast as the denominator?   