# How to Find the Horizontal Asymptote (NancyPi)

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Published on May 15, 2018 by

MIT grad shows how to find the horizontal asymptote (of a rational function) with a quick and easy rule. Nancy formerly of MathBFF explains the steps.

For how to find VERTICAL asymptotes instead, jump to the video: https://youtu.be/V137qmDN9Qw

Follow me on Instagram! https://instagram.com/mathnancy
Follow me on Twitter: https://twitter.com/nancypi

The degree of a function is the highest power of x that appears in the polynomial. To find the horizontal asymptote, there are three easy cases. 1) If the degree of the numerator expression is less than the degree of the denominator expression, then the horizontal asymptote is y=0 (the x-axis). 2) If the degree of the numerator is equal to the degree of the denominator, then you can find the horizontal asymptote by dividing the first, highest term of the numerator by the first, highest term of the denominator. This will simplify to y = some constant (just a number). 3) If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.

For more of my math videos, check out: http://nancypi.com

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### 209 Comments

• It would be of ur great help if u could explain the chain rule.. Please..
Student from INDIA.

• PTA: power, trig, argument. Differentiate each part with the rest untouched and multiply those results.

• DV Thank u 😊😊

• She has some calculus videos on her old channel. I imagine they’ll migrate here eventually.

• Don’t stay up too late.

• Haha, 5am already. I’ll take note of this next time…….

• I don’t want to, but the test is tomorrow!

• clear

• clear thanks 👍👍👍

• You are the best Nancy! 😉

• I can’t maths at this hour of the night, but I’ll check these out later

• Maths 24/7 brah.

• @Math and Logic totally brah. 24/7/365

• But what if we wanna mess with limits? Does the proof involve limits? Are we just letting x go to infinity?

• Yes, taking the limit to pos. or neg. infinity is equivalent to finding horizontal asymptotes.

• This is really for Algebra Students. For those who have not touched the studies of calculus. She says do not mess with limits not to stop the knowledge of limits but to prevent unnecessary confusion.

• DavidOm 0221 I get that. This is review for me, but it makes a lot more sense now. It seemed completely arbitrary at the time and I forgot the rule again already so it helps me to try to make sense of it all.

• @DavidOm 0221 That’s why I stuck to 2 dimensions in my geodesic equation derivation. Sometimes I’m willing to sacrifice total accuracy for clarity, even though it’s usually very frowned upon.

• Nice video 👌👏

• Happy to Learn from You

• can you do one for Vertical Asymptote

• vertical asymptote is setting the denominator of a polynomial to = 0. then just solve for x

• Unless the numerator has that same zero. You need the rational function to be in lowest terms for your solution to be valid.

• i dont even have math to learn or do, i just watch because of nancy….

• I guess the second hand smoke method of learning is better than nothing.

• Hey Nancy
You look very sad and tired
What happened?

• Ya , i also fell it, namcy take your time. We all here for you

• love the quick rules that you used for HA’s!

• I love your videos <3

• Hi u r beautiful and plz make more videos in calculus

• You should tutor people online

• Hi Nancy do u use what’s app 9825783397 i love the way u explain ..chat with me or also check NIKUL K BHATI on face book

• Nancyyyyy♡

• She said DONT MESS WITH LIMITS with such devilish eyes 😂

• true dat xD

• Priest nanci

• how are you writing in the air

• In case the degree of the top is higher than the degree of the bottom , you have a slend asymptoot , ONLY when the upper degree = bottom degree + 1 . The long division gives you a a line . When the difference is higher than 1 , you will not have a slend asymptoot

• In all those cases, if the top is of higher degree, then the quotient you get from long division is the (end behavior) asymptote. For example, if you did the long division and you got a quotient of degree 2, then the end behavior asymptote would be that 2nd degree polynomial. Horizontal and slant asymptotes are for linear end behavior asymptotes, but for higher degree asymptotes, they haven’t given them names. So one could say “quadratic asymptote” to indicate a 2nd degree asymptote, or “cubic asymptote” for a 3rd degree asymptote – go on from there.

Whatever the end behavior asymptote, if you let your graphing calculator graph the original function together with the asymptote and zoom out enough, the graphs would seem to be identical; the idea being that for large values of x, the function and its asymptote are pretty much the same.

• Saved my life! Thanks.

• Hey I am a student in Botswana working on a project to help students my email is tbadzili@gmail.com please send me yours so I can give you the full review.

• for top degree, it’s none, but a more formal and exact way to write it is infinity.

• But I do like math!!!

• Such a great help for our exam tomorrow! I love how u write and explain. Thank you! ❤

• You’re videos are always helpful. You’re an amazing teacher

• mam plz give a detailed explanation of horizantal asymptotes, its origin, related concepts , where they are used and some examples if u explain them u r the one will become the person who helps me a lot. waiting for answer . plz post it soon or make a video on it

• for engineering level first year is this much enough

• Your tip to find horizontal asymptote really helped me. Now I don’t have to get to the standard asymptote form.

• Superb Nancy

• so, what is this rule call?

• This kicked ass

• Thanks to this video i made my homework 😂

• You saved me thank you 🤧

• Thank you! I was on the verge of smashing my head on my bed out of frustration

• Thanks!!!!

• DONT MISS WITH LIMITS

• Plz post more videos
Love your teaching Nancy

• Nancy, why is it that you can take a simple concept and make it simple, while my professor takes a simple concept and make it extremely difficult. IT’S BECAUSE MY PROFESSOR WANTS TO SOUND SMART!!! You, on the other hand are humble and good. You don’t want to impress anyone, you just love math. So beautiful!!! Inside and out!!! YOU!!!

• It was very clear! Thank you!

• Isn’t (x-1)/(x^(3)+1) for x = 1 a contradiction to: “top degree < bottom degree -> H.A. is y = 0″?… What am I looking at differently here?? Thanks!

• Ok like great trick, but can u explain Y that works, I’ll get fucked if I have to explain why or do an application question with it.

• You remind me of Violet from the Incredibles

• what if theres no powers?

• great as usual. lighting is kinda bad here but no problems ur other vids are fine

• You don’t have a video for slant/oblique asymptotes? 😫

• Stop making things this easy it is starting to scare me

• It shows your learning if it looks easy to you!!! It was hard before right? Now you truly understand

• SIMP

• @Nico Piazza ….

• @Nico Piazza you’re so cringe

• you are amazing!! 😱i cant believe someone made it so simple…. now i look at my class notes and wonder if my teacher made it so complicated on purpose🤔

• for real. I feel like he wants to make it harder to seem smarter

• would you take me to prom?

• i like how u write opposite

• Awesome videos!

• What if the rational function is factorable, do I do that before solving for HA, because then I would have 2 factors which I’m stuck on

• To my knowledge you would not factor first. You would take the degree given.

• Brandon Beins so would I simply the rational function then foil it to find the highest degree and just leave it as that to find HA?

• such HQ work- great job Nancy.

• Hi nancy

• this video saved my life

• Yo somebody feed nancy

• Clicked on it when I saw MIT grad xD

• This video makes H.A. so much easier to solve. Thanks!

• How in the hell are you writing backwards????

• @Waqqi Ali i know im just trolling

• @Waqqi Ali well, she writes on glass just like you and me write on paper, records it from the other side of the glass and the uses a software to mirror the whole video so that we can read it.

• @Thejas Kodoth Wow, I didn’t see that, thanks

• She is wizard Harry!!!

• @Detiz Dynamite no that’s impossible then she’d still be writing weird i think

• I love you Nancy Pi💚❤️, thank you and thanks god for running into your videos , keep going your helping so muchhh

• I cannot believe it is that simple! Thank you so much!

Also, you are very skillful to be able to write backwards that well. 👍

• shes so smart she can do this and write backwards!

• You are fantastic thanks a lot.

• Thanks, NancyPi, you make Math so easy for me. Good job

• you look like a robot

• How are u able to write backwards, that’s crazy. And u write so neat and straight, mind blown.

• you got the scariest eyes

• You deserve a Nobel prize since you help the students overcome one of biggest rock in the way pursue a career or major; Maths.
You make Maths seem so simple. God bless you, Nancy for doing a good job

• Thank you so much you make maths so easy

• I hope you record theses writing normals and then mirror the image. If not you are possessed. But hey helpful video thanks!!!

• Thanks ms. Nancy my final exam about 3 hours later and you give me a huge favor by explane me this semple rules peace🌹👍

• OMG! I have a quiz tmw. And I’m freaking out, but you made it so easy!!!!

• Wow, my textbook and professor made it so complicated. I love your video and you are excellent! I’m so glad I found you, may you have a successful and bright future!

• The video is inverted. Your welcome

• wow. beautiful, smart, and explains it clear and concise so i am able to understand. subbed! thank you.

• Slant asymptote ??

• You’re so beautiful and smart c:

• her voice gives me ASMR ha ha:)

• Weird flex, but okay

• Thank you very much. please keep up doing more videos. Really appreciated!!

• Okay but asides from the fact that she’s a great teacher she’s now reached the top tiers of awesomeness because of her ability to write backwards so perfectly

• You make it so easier to me thanks a lot 😍

• what is this h.a
e^(x-1)/(x-3)
@nancypi thanks

• Thank you so much

• what if you don’t have a x on the numerator and just have it on the denomintor what do you do then

• But what is a horizontal asymptote?🤔

• Thank you!

• People really came here to learn about asymptotes and are still confused on how she rights backwards

I’m really confused right now

• But oblique asymptotes? These are the hardest

• Your way of tackling math problems is just amazing. Love you so much. You make math easy for me

• Awesome. Thank you so much

• Yay, thank you!

• thank you . you made my day : )

• Nancy look tired in the vid , and tyyy I understood now 😘

• What the fudge. I know every student is different and learns differently but gosh I wish I had this kind of teaching in high school. I’m here struggling in college algebra … in college. I have my exam tomorrow and youe videos are helping me so much. I wanna cry lol. Omg God bless you! Thank you!!! 🙌🏽

• you look tired, don’t do too much hard work.

• Slant asymptote?

• Thank you Nancy

• You fine 😭

• THANKYOU

• how do you make it so easy? Thank you so much, if it weren’t for your videos I’d be failing!

• This don’t make no sense my equation looks a lot more different

• simplify ur equation dude

• This was three days ago lol

• thank you so much. you are a genius!

• thank u! u explained it so much quicker and easier than my textbook

• i love u

• Thanks a lot.

• NAAAAAAANCY

• What if the numerator is inside a square root?

• That’s why you didn’t write anything in the thumbnail because it’s none which means invisible.

• Thanks alot, books and teachers tend to over explain such a simple concept.

• I don’t wanna mess with limits so i liked your video 🙂 Cheers Nancy !

• Thank you madam I didnt know how to study the horizontal asymptotes

• Nancy Pi is my HERO!!!!!!!

• Great videos. Multi Color and Writing backwards is amazing. Oblique Asymptotes Please

• I’ve been wondering all through my math classes, “there has got to be an easier way to learn these things”. Thankfully, youtube made it possible for us students to see those ppl who found a simpler way to explain math!

• NancyPi saving my math grade AGAIN 😀

• all i have to say is thank you for making this soooo simple:)

• Literally have an assignment due in the morning and u saved my guts tysm:)

• gracias fue de gran ayuda,se me abia olbidado

• asympototes of
x2/x-1

• A-symptote with a long A as in asympomatic or atypical, from greek a “not” + symptote “coming together, rather than “ass’m tote”.

• lol the ads at the start of these videos always pander to anxious students with bad instructors… like get outta my face, NancyPi is about to teach me exactly what I searched for

• Thank you!

• you made me love math and have a dream to teach it to my learners to be the way you do it. stay cool

• Thank you so much.Your 5 min video is way more better then couple of months in math function class.

• Pretty and SMARTTTTT

• Thank you Nancy, passed the test with full mark.

• is horizontal asymptote the same as end behavior?

• Thank you. You’re the best.

• Ha ha ha

• Your videos are amazing, I have to take my algebra final on monday and i know im going to pass it because you explain these concepts in a way I can understand them.

• thanks a lot

• thank you SO much for this. i’ve been crying all morning not understanding horizontal asymptotes and this has literally saved my life

• If i get an A on my Calculus test can i go on a date with you?

• Hey you make a cute hobo.

• are you writing backwards? o.O
holy hell…

• i wish my skin looked this good

• Thank you!

• Thank you for your teachings.

• Awesome 🙂 thanks

• Nancy, you are beautifully smart

• thanks nancy

• Haha “you dont have to like math but u can like my video”

• Can u make a video about oblique/slant please???

• this is the best video for this subject . u’re the best teacher i had ever seen.

• Very excellent explanation

• I have been seriously stressing over this stuff. Your videos have made this extremely easy to follow and understand!

• Damit, i learn so many thing from Nancy and its funny how most of the thing that u thought it hard, it actually very easy🤣
I learned Limit, Verticle, horizontal asymptote and derivative from u so far. Limit and derivative, i havent reach those lesson yet, but i alr know how to do it all by watching those 10-15mn videos. Its just so “work” that i find it funny. 🤣

• blesss

• Not true for y=(-x^2 +6x +2) divided by (2x^2 +14x +12) which should have a y asymptote of -1/2, unless you are saying you ignore negative symbols, than true for y=1/2…still confusing

• JUST SAVED ME ON THIS TEST THANK YOU

• Thank you Nancy

• sup boys

• is not she the cutest?

• Violet from increíbles

• thanks for the video. I tried to understand what my book said about horizontal asymptotic, but they can’t present it in a way easy to understand. they like to complicate everything

• Thank you Nancy! You’re the best

• ahh.. thank you for making this so simple 😭

• you saved my life during quarantine. if it wasn’t for you, i’d fail algebra

• Nancy, your videos have assisted me greatly in my Business Calculus Class. You are able to break down the concept and problem in such an incredible way that people like myself who have struggled all my life to understand Mathematics can learn problems I never thought imaginable. Thank you!

• what does it look like on a graph…. that would help thanks

• So basically, it is what is called ‘limits’, isn’t it

• Good Work. Learning made easy. Thanks.

• how is everything this easy, what is my college professor being paid for? making me watch nancy do his job?!

• i have failed one exam because i got these wrong, i wish i could go back, you have made it so easy

• can someone PLEASE tell me how she’s writing from behind the glass but still the letters aren’t flipped

• She writes on a glass in front of her, and then edits the video so the image flips horizontally. She is actually right handed.

• excellent teacher Nancy, you earned more one subscribe.

• Great explanation!!! I am still going to practice the oblique or slate asymptote. Can you make a video on long division with equations?

• Thank you ma’am 😄

• Whyyy did my lecturer teach me limits and not this simple rule 🤦🏾‍♂️

• I have a visual representation as to the three rules why they are the way they are:

1) top degree>bottom degree: in this case, as x gets arbitrarily large (which always happens on an infinitely large plane), the quotient (x/x2) becomes lightheaded, on other words, when approaching large values of x, the x squared on the bottom is always larger than the top, so it is approaching a 1/∞ case (an infinitesimal) that’s why the graph approaches 0 but not quite zero, same as the property of 1/∞.
2) second case x2/3×2, same degree top and bottom, any term that is not the highest degree will be negligible in value once x reaches near ∞ (which, large x values will always be present on an infinitely big number line). So in this case, the leading coefficient matters a bunch, because as x gets arbitrarily large, the ratio of the large leading terms will be determined by the leading coefficients (∞/3∞) which is 1/3, but not quite, so it approaches horizontal lines in the ratio of the top and the bottom leading coefficients.

3) top being greater than bottom degree, this case, the bottom will get smaller the larger x values you put in. And again, any non highest degree terms will be negligible with high enough numbers, like adding 1 to 10000 is 10001, but approximated ten thousand, just as (10000+1)/(100+3) is 10001/103, approximately ~100. But as x gets larger, the gap between the top number and lower number increases because of the degree difference, so the value approaches infinity. For this, there is not defined asymptote, because the asymptote is located at y=∞, where the hyperbolas are gradually approaching. ( the asymptote exists but we can never see it, it’s ‘so high up’ that it’s undefined).

• Nancy is 10/10

• I absolutely love how u make it so understandable ❤️💕💕❤️and so simple I absolutely love you 💖

• I was not taught this and became confused in my final review. This ideas seems to be the only one I could find that didn’t mess about and just explained it. By far the best video on this topic.

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