Horizontal Asymptotes and Slant Asymptotes of Rational Functions

This algebra video tutorial explains how to identify the horizontal asymptotes and slant asymptotes of rational functions by comparing the degree of the numerator with the degree of the denominator of the rational expression. The equation of the slant asymptote can be determine using long division if the degree of the numerator exceeds the degree of the denominator by exactly 1. This algebra video tutorial contains plenty of examples and practice problems.
Fundamental Theorem of Algebra:
• Fundamental Theorem of...
Rational Expressions - Basic Intro:
• Rational Expressions -...
Simplifying Rational Expressions:
• Simplifying Rational E...
Multiplying Rational Expressions:
• Multiplying Rational E...
Dividing Rational Expressions:
• Dividing Rational Expr...
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Adding & Subtracting Rational Expressions:
• Adding and Subtracting...
Rational Expressions - Unlike Denominators:
• Adding and Subtracting...
Simplifying Complex Rational Expressions:
• Simplifying Complex Ra...
How To Solve Rational Equations:
• Solving Rational Equat...
Rational Equations - Extraneous Solutions:
• Extraneous Solutions o...
_____________________________________
Horizontal and Slant Asymptotes:
• Horizontal Asymptotes ...
Finding Rational Functions Given 2 Points:
• How To Find a Rational...
Rational Functions - X and Y Intercepts:
• How To Find The X and ...
Graphing Advanced Rational Functions:
• Graphing Advanced Rati...
Rational Inequalities:
• Rational Inequalities
________________________________________
Final Exams and Video Playlists:
www.video-tuto...
Full-Length Videos and Worksheets:
/ collections

[@TheOrganicChemistryTutor]

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I have a teacher for calculus 1 that hasn't taught for 14 years and came back and she is terrible at teaching she goes on the textbook says copy down this example (explains nothing) then says to do questions 1-4 without telling us how to do it so every class we have to learn it on our own

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[@whipzy4301]

very late reply but there is a quicker way called synthetic division, when you're dividing by those trinomials with a binomial it makes it so much easier.

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10:01 lmao i thought it was (7x-y) + y but the y was actually 4 😂 anyways great video man!

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[@The_jezeb3l]

tenure this is when school sign contracts with teacher pretty much guarantees them a job for life no matter what. That's why when you get a bad teacher most of the time they get away with it. Even if they are so bad and the school decides to fire them the school has to go through a complicated process that can last at least a year or longer. Chances of a teacher getting fired is 1 in 2000 compared to a doctor which is 1 in 57. The American public school system sucks. We need more teachers like this dude.

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MR. Organic Chemistry tutor ,thank you for a well explained and solid analysis of Horizontal and Slant Asymptotes in precalculus.

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[@taiedy]

Sorry, how can there be a slant asymptote if the equation at 5:20 factors cleanly into [x+2] without a remainder? Shouldn't there be a slant asymptote only if it leaves a remainder?

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[@sonjahenderson1390]

Im confused about if these rules ALWAYS apply. For instance take (5x + 1) / (x^2 - x - 1) and graph it in the graphing calculator. There appears to be a horizontal asymptote at y = 0. However there is a vertical like that strikes through y = 0 in between 2 other lines in the function. Does the horizontal asymptote still apply?? Do these set of rules apply no matter what? If anyone could help that would be wonderful!! Thank you

[@nejcocepek8357]

in the first case of slant asymptote the asymptote does not exist since the remainder of long division is zero and if we cancel out the terms we see that we get linear function with equation y = x + 2 with a missing point at x = -3 which lies on the asymptote and that contradicts with the definition of an asymptote

[@ad-np8ko]

exactly i was confused on why he said it was slant

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[@noeldiaz1354]

its easy to remember if u think like this... Think the rational expression as a building if its top heavy just by one it will collapse but not desintagrate resulting in a oblique line if its top heavy by more than one it will collapse and desintagrate, theres no line! if its bottom heavy well the building its on the floor as it should be, y=0 if top and bottom are equally heavy, then divide those bad boys

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[@mr.9754]

Dude....(x squared + 5x + 6)/(x+3) = (x+2) which is a straight line with a hole at x=-3 i.e. There is no oblique asymptote in that example. ps You should have seen this as soon as you realized there is no remainder.

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[@mr.9754]

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[@kfjfkeofitorhf9520]

24X^2---6X^2==y 24---------6=====y 18/X^2+3= 18(X^2+3) 18X^2+54 X^2=54/18=3 X^2=3 X=-1 --8'-6/--1+3= -14/2==--7 X=1 8-6/4= 2/4= 1/2

[@bobu5213]

is there a way to rationally calculate these without knowing any tricks like "If the degree is the same devide the coefficients". I want to know how to look at a function and see in my head where the asymptotes would be

[@barrym5310]

By the way, synthetic division can be used to find the asymptote, although it might be easier to use polynomial long division in certain situations.

[@daniaalsadi7208]

Is synthetic devision the same as long division?

[@sauffle]

@@daniaalsadi7208 you end up getting the same answer in both but the way you get that answer is little different.

[@qarashuaq]

imo synthetic division is more convenient to use

[@jinnjinn5567]

4:00 - isn't this just equivalent to x+2, a linear function with no asymptotes?

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[@philipversteegh1033]

y=(8x-6)/(x^3+3x) its not just a x=0 asymptotes x=-3 is also one

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[@jackwilson4324]

I'm talking about: x^2+5x+6/X+3, but algebraic long division is probably something to get marks in the exam, rather than factoring, both ways are not that hard, after you explained them.

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THANK YOU 💚👀✨ I didn't understand how to find Slant asymptote, but I'll understand I watched this video twice, and now I better understand how to find slant asymptote :з)

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Someone please answer my question,why do horizontal and oblique asymptotes exist?

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I know how to solve the questions, but what is the logic behind the question?

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[@jstagirlinthisworld]

So, there is a major maths error in this video. Starting at 5:00 We are given the equation: (x² + 5x + 6) y = --------------------- (x + 3) which, when properly factored, is: (x + 2)(x + 3) y = ----------------------- (x + 3) So, even though the numerator is 1 degree larger than the denominator, it also shares a common factor with the denominator of (x + 3). Before dealing with asymptotes, we have to remove any *holes* that may be present in the graph, and this matching (x + 3) factor tells us that there is actually a HOLE at x = -3, not a slant asymptote. Removing this hole from our equation gives us our actual graph which is now reduced to just the graph of: y = (x +2) with y's domain restricted to all real numbers *except* x = -3 that is, our function is simply: y = x + 2 Domain: (-∞, -3) ∪ (-3, ∞) Plug this into a graphing calculator if you want to double check - the graph is just a straight line that is undefined at x = -3 You can't have an oblique asymptote if your graph itself is just a line

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[@The_jezeb3l]

*Why do bad teachers get paid?:* tenure this is when school sign contracts with teacher pretty much guarantees them a job for life no matter what. That's why when you get a bad teacher most of the time they get away with it. Even if they are so bad and the school decides to fire them the school has to go through a complicated process that can last at least a year or longer. Chances of a teacher getting fired is 1 in 2000 compared to a doctor which is 1 in 57. The American public school system sucks. We need more teachers like this dude.

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I thought if the degree of the numerator is greater than the degree of denominator is simply no asymptote?😭😭

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The third example at the end.. why is it bottom heavy? Wouldn't they both have a degree of 1? 10:05

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Perhaps I am just missing something, but on the problem, 2x^-8x+6 /x^2+4 , how did you he get -16x in the long division, I thought it was supposed to be -8x which would make it -8x after the (-) is distributed inside the parenthesis.

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Why there is no slant asymptote in y = (x^2 - 3x -4)/(x+1)? According to the video, there should be a slant asymptote!

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Didn’t need long division for the first slant asymptote since you could factor the numerator which cancels with the denominator, leaving x+2.

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[@hinauzair7121]

Is slant assymptote is same as oblique assymptote

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the oblique asymptote is just the quotient/function of the division doesnt have to be only 1 higher

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7:54 I wish my teacher taught us this before the quiz. I was freaking out because of the remainder since the mulitple choice answer didn't contain the remainder.

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I don't understand why when its bottom heavy then y=0. Where can I find the other video where you explained this?

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How do I find a teacher who analyzes this without division

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Thank you so much I have an exam coming up for pre calc and this low key confused me until u explained

[@FunComputerScience]

what if function involves trig function? such as 11x^2/(x+1)+sinx

[@GSU_Panther_Nation]

I always find that synthetic division is much easier than polynomial long division

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What about the Vertical asymptotes for the very fist discussed examples

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