# Vertical Asymptotes… How? (NancyPi)

178,320
Published on May 17, 2018 by

MIT grad shows how to find the vertical asymptotes of a rational function and what they look like on a graph. To skip ahead: 1) For the STEPS TO FIND THE VERTICAL ASYMPTOTE(S) and an example with two vertical asymptotes, skip to 0:19. 2) For an example in which FACTORS CANCEL and that has one vertical asymptote and a HOLE, skip to 5:58. 3) For an example with NO VERTICAL ASYMPTOTES, skip to time 10:12. Nancy formerly of MathBFF explains the steps.

For how to find the DOMAIN of a function jump to: https://youtu.be/GQGFMUfr10M

What is a vertical asymptote? It's an invisible vertical line that a function gets really really close to but never reaches. How do you find the vertical asymptote(s) from the given equation?

THREE STEPS TO FIND THE VERTICAL ASYMPTOTE(S): For a rational function, there are three main steps you can always follow to find all the vertical asymptotes, if there are any:

STEP 1) FACTOR: The first step is to factor the top and bottom (numerator and denominator) if you can, and as much as you can. For instance, in the function f(x) = (x^2 + 3x - 10)/(x^2 - 4), you can factor both the top and bottom. The numerator, x^2 + 3x - 10, is a quadratic that factors into (x + 5)(x - 2), and the denominator, x^2 - 4, is a difference of squares that factors into (x + 2)(x - 2). You then rewrite the whole function with both of these factorizations so that you have f(x) = [(x + 5)(x - 2)] / [(x + 2)(x - 2)].

STEP 2) CANCEL: Next, simplify the function by canceling any factors that are the same on top and bottom. If there are no common factors, you can leave it alone. In our example from Step 1, there is an x - 2 term on both the top and bottom, so we can cancel those two factors. You can rewrite the function after getting rid of those similar factors so that it looks like: f(x) = (x + 5)/(x + 2).

STEP 3) SET THE DENOMINATOR EQUAL TO ZERO: After simplifying and getting rid of any common factors, the last step is to find the real zeros of the denominator by taking the bottom of the simplified function and setting it equal to zero. You then solve that equation for x, and any real numbers you get as a solution for x are where there are vertical asymptotes. You can write your answers as just "x equals [some number]". If you have vertical asymptotes, they will always be in that form, such as x = 3 or x = -2. These represent vertical (invisible) lines on the graph that your function approaches but never crosses.

Remember that if you get an imaginary answer when you solve for x (such as a square root of a negative number), then there are no vertical asymptotes. If there is no real solution when you solve for x, then there are NO VERTICAL ASYMPTOTES. Note: By the way, if you had factors that cancelled in Step 2, that created a "hole", or removable discontinuity, on the graph where the function was indeterminate.

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

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• Her titles be like…
*Verticle Asymptotes… WTF*

• Abhishek Patil My students usually like my jokes, but whatever. Can’t please everybody.

• ๐

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• i thought it was kinda funny

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• ๐๐๐๐๐can u please even make a video how you got into mit i m very inspired by the way i love vertical asymptote

• Ankur Giri all about practice and love

• I already know this, watched for her

• my bois a scholar

• Happy to see you back again

• VERY happy to see her ๐

• Hi Nancy,you are beautiful, plz make more videos in calculus…check Facebook ‘nikulkbhati’

• From Indonesia. Hebat, semangat Nanci

• It would be cool if you had advanced math videos too.

• Lucid way of teaching! love from ๐ฎ๐ณ

• Can you post videos on integration

• TECHNICAL ADDA she didd in her previous channel Math Bff

youre welcome.

• Hi Nancy! Could we please get a video for partial fractions?

• Your math videos are really good quality. Excellent job.

• Finding asymptotes is so much fun. Graphing them, oh my goodness yes!

• Unpaid Cringy Advertisement: Before I started watching NancyPi, I didnt know my asymptote from a hole in the graph. Now I wipe math problems all day long. Thanks Nancy! (cheesy smile).

On a serious note. Excellent, clear, easy to understand videos. Thank you.

• I know this topic but I am here for ASMR

• Kk omg yes!!! Her voice is so soothing! โบ๏ธ I actually have a math exam in the morning (8am) and now itโs 1am and her voice is making me so sleepy I have to stop studying and go to bed. ๐คฃ

• yeah math tutorials are great for it

• ASMR?

• Thanks for teaching me how to find the vertical asymptotes of a rational function.

• I missed you ๐

• awesome, great video

• Nancy,

F(x)=(x+1)/(x+1)(x+1)

• You cannot remove x-2 from the function ! Like you explain , x= 2 is a hole and (x+5)/(x+2) does not have a hole x= 2

• As always thanks for taking the time to do these videos. Great quality and easy to understand, well done. Glad to see you are back.

• lim x {x:–>(1/0)}
A feeling I have missed for decades.
Thank you.

• thanks…..ur so sweet….

• I thought this would be hell, but you have a good way of calming down peoples nerves. You know math is hell, thank you for reducing it to hell-2x^2

• she is sooooo relaxing

• I think its fun

• Thanks Nancy!
I’ve watched a few of your videos to help catch onto some basic stuff for college algebra quickly. You are a natural educator! Love the see-thru board for visual reference and your approach to the material. You know how to make people watching feel like “relax… we got this!”
Please keep posting. I plan on getting all my math knocked out with your videos! LOL

• You are so relaxing!! But don’t get confused about your viewers’ thoughts!

• I have a great respect for you you , you save my life

• it’s so nice to see you again nancyyyy

• ph

• ur hot

• Thanks A Lot U Dont know How Much U Helped

• Question – When does the infinity or negative infinity symbol apply??

• Is she writing the opposite way?

• the video is flipped horizontally

• i’ve been away from the maths for a couple years but now have to wade through calc. SO thankful that i found you again!!!

• You FREAKING ROCK!

• I love u

• Damn I get sidetracked easily. this whole time Iโve been trying to figure out what type of sorcery is this?! Are you writing inverted or backwards? Hmm is the camera inverted and your really writing with your right hand instead of your left ??๐ค maybe she edits it ?? So many question, only getting VA answers.

• I know my response is pretty late but here you go

• @T0XIDITY holy shit thank you ๐

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• joe manuel lol no problem lol๐

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• Thank you so much!! Youโve helped me so much through Calculus & Pre Calc! You explain everything so well. Thank you thank you โค๏ธโค๏ธโค๏ธ

• so I did the example on my own just to see if I really knew how to do it & I factored the denominator into (x+6)(x-1) help me ๐คง

• if you foil this, you times 6 by -1 which equals -6. In the polynomial, the 6 is a positive.

• She said hi and I instantly liked the video

• i dont know what i would do without you. Dr-teachers in uni cant explain a simple thing when you explain the whole thing in couple of mins when they take an hour and 50 mins for a single idea and still people confuse to solve them lol.

• So is the VA just the values the domain of the function doesn’t include

• hot white chick

• What do you think shes saying at 9:18

• thank u

• ูุณุณุณุนุฏ ุงููู ุงูุง ุจุฏู ุงููู ุงููุงููููุงุณ ูู ูุฑุงูู โคโคโค๐๐๐๐

• E. yousef AL-Aga what

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• can’t focus on the lecture. She is distracting!

• My only regret is that I did not discover this channel earlier. You are making calculus way too easy. Thank you for saving my grade.

• he he, slight error with the plot of f(x)=(x-1)/(x^2+5x+6), at f(0)=0 but at as x->+infinity, f(x) -> +0 from the positive side, so the graph goes up and though the origin and then comes down and approches the x axis as the horizontal asymptote. Whacky graph!

• HOW CAN I FIND OUT THE DIFFERENT ASYMPTOTE EQUATIONS FROM A GRAPH? WITHOUT HAVING ANY INFORMATION ABUT THE FUNCTION EQUATION?! PLEASE!

• NANCYPI…. YOU ARE JUST AMAZING!!!! MY FIRST VIDEO WAS INTEGRALS…… FROM PERU, I JUST WANT TO SAY: THANK YOU VERY MUCH!! NEW SUBSCRIBER HERE. HOPE YOU WILL KEEP POSTING REALLY USEFUL VIDEOS….. L.I.A.T.E <3 REALLY, FROM MY HEART, I JUST WANT TO THANK YOU :') REGARDS FROM PONTIFICAL CATHOLIC UNIVERSITY OF PERU

• May I put My Vertical Asymptote in your Horizontal Asymptote

• thanks Nancy ๐ฉ๐ฉ๐ฉ

• 9:17 funniest word twister i replayed it like 20 times. Great step by step process

• โค

• Making my math world a lot easier to navigate. I love you Nancy!

• Thanks, very helpful video.But if the denominator is just “X” value in a rational function, how can I find the vertical asymptotes .

• x would simply be zero, I think.

• Wow! Unlike most math teachers, this one actually speaks like a human being and explains stuff in the way a human being can understand.
Thank you.

• I know right!!! It’s like you going to a doctor and they say all this medical language to you for some time, and then you have to interrupt and say, “In English, please.”

• But the way she writes is weird, inverse, and creepy. Despite for giving me creeps, she also gives me great knowledge that motivates me to become an engineer.

• Wow, amazing. Even other youtube videos could not make it this clear. Also, I get a vibe that you are a human being on the side, I enjoy that too, it makes me feel less desperate for some reason.

• Thank you mam.It was really helpful.

• thx for every thing

• Ur better than my own math teacher thnxxx

• Y’all are all here to creepily hit on this woman who’s just teaching im just here for the math dawg im tryna pass this test

• Y’all here to deny the biological mechanisms present in nearly all species.
1. Female displays availability or not
2. Males demonstrate interest
3. Males compete
4. Female chooses based on competition or competitive factors.

• @MrAcebak Just because its a math video doesn’t mean you cant watch it without doing math. Its on the internet and people watch it.

• @MinerBoB I think you missed my point

• +iionic stfu

• Very nicely explained, as always, Nancy. Crystal clear. And thank you for always being so personable and sweet in your presentations.

• ๐

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• Maybe the helicopter when the engine turn on
Are this coantety vertical asymptotic

• Horizontal lines

• Dinmometr + horizontal +asymptotic = vertical no show byonic speed

• How are we supposed to concentrate when ur so hot๐๐ฅ๐ฅ

• Please do oblique asymptotes too! I had to go to PatrickJMT figure them out. Thanks for all this though. I’m doing well in my Calc I course thanks to you!

• In example 2, is it a curiosity that the reduced form “x+5/x+2” is wow ๐magically now defined at x = 2 ….by virtue of just cancelling the x-2 terms top and bottom? …..the inequality must be because f(x) at x=2 has been negated.

• Mam how to we Find Asymptotes of transcendental Functions?

• did u attend to MIT and why r u so beautiful?

• Nancy, You are not only lovely but your videos are very helpful and you explain things simply. Thank you for doing what you do.

• She is ๐

• Lovely

• fancynancy

• For the first example is the vertical asymptotes infinite or removable?

• Hard to concentrate when youโre this hot nancy

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• Thank you that was really helpful

• Can I just have NancyPi from now on please?? Thank you.

• A calculus goddess

• NancyPi i really love your step by step approach to this math problem. God bless you. But i have a problem with the question two. The answer you got was -2 and that was dependent on the factors you chose, i.e, +5 and -2. I on the other hand chose -5 and +2 as my factors and as such the answer is +2. Does that mean my answer is wrong??? Please get back to me.

• The order of your signs matter. Because you have to look at the original equation. Even though you got the numbers right, if you put your sign in the wrong place, it will screw up your answer. When you foil out the factors, you should get your original quadratic. If you do not have the same signs as the original, then you put them in the wrong place.
EX: (x+5)(x-2) These are the correct factors. Why?
FOIL = First, Outer, Inner, Last
First term: x^2
Outer Term: -2x
Inner Term: 5x
Last: -10

Ergo when you combine like terms, 5x-2x = +3x
Therefore your signs match the original. We have a postivite x^2 a positive 3x, and a negative 10.

What you did would have been right if the signs of the original quadratic were -3 and +10. Does that make sense? I really am not good with math, but I hope I helped you somewhat.

• @Hyujabut44 Thank you very much. You are right and I understood you. God bless you too.

• she needs a joint

• I found your trick you simply flip the video.๐๐๐

• Thats not trick man

• Just simple logic

• As I checked, it seems both of your graphs are wrong. please check it on Grapher or Mathcad.

• Thank you so much, I checked out a lot of vids but this one’s the clearest ๐

• Easy..

• Great video, reviewing for finals. Thanks!

• Doubt cleared….Thank You… Nancy Pi

• You can factor x^2+1 with imaginary #’s. (just FYI)

• Is that limit ?

• You said everything except what I fing needed, you teachers always forget something, what if you have x squared+9/x squared – 2x – 15. You teachers always miss something and never fing say it. And when. Some one like me is searching for it you guys never cover it!
If your going to teach cover every angle of an equation, is stupid how yous guys do that, its frustrating! COVER EVERY ANGLE! Please

• Thank you

• Thanks nancy๐

• Your method of delivering the lectures is superb

• damn i cant focus cuz ur so beautiful

• Hi Nancy. under what conditions can we find the vertical asymptotes? does the degree of the numerator has to be smaller? can we still find the vertical asymptote if the degree of the numerator is greater than the degree of the denominator??? please!!!

• A good exercise would be: Find V.A of the following functions: y = tan(x) and f(x)=e^x+e^{-x}. This are not rational functions so….. how do you determine the V.A in this cases?

• I have a doubt, when we have cancelled x-2 from the function then why it is not defined at x=2

• because then, it means that an x-value does meet 2 where as in a vertical asymptote, it means that it never touches that number. if i’m not mistaken.

• @reese ok
Thanks

• Simple, clear and precise, Math is a lesson in logic. Tank you, Nancy.

• Saved me tears.
My teacher taught this yesterday and now wants us to know it by heart for tomorrow. You saved my grade.
Kudos!!

• She is the math queen!!!!

• focus the right side of your screen! don’t forget you are learning math here , not watching girl.

• Hilarious … i am trying

• ูุนุทูู ุงูู ุงูู ุนุงููุฉ ุนูู ุงูุดุฑุญ ูุชุงุจุนู ููkfupm ุงุญุณู ูู 100ุฏูุชูุฑ ูุฌุชูุนูู

• @Nursin 1999 ุงุฏุฑู ุจุณ ูู ุจุงุจ ุงูุชููุน ูู ุงููุบุฉ ูู ุงูุชุนูููุงุช

• Hottest math major

• Nice videos! very well explained!!

• Keep up the good work!

• This is a lot easier than graphing calculus concavity. I’m still struggling with that… 8c

• 4:30 to 5:00 One correction, you say the functions never cross the asymptotes which is true, but they actually will never REACH them. The difference between x < 2 and x <=2.

• I think she mentioned that at another time in the video, I can’t remember where though.

• I love her channel. Because she is pretty I do not get bored of the video and I can focus on learning if that makes sense lol.

• Thank you, Nancy, excellent explanation like always. God bless you.

• those littlel moments where she writes in fast forward, imagine how much clock time it takes to edit that in.

• Maybe just 5 mins

• How tf does she write like that

• thanks a lot sir

• you are great

• if only my math teacher looked like this

• wow Im seriously embarrassed that guys cant seem to keep in in their pants on a flipping math video… Im ashamed of my own gender yet again. nobody comments this sort of shit on math tutorials with guys instructing

• Unique ๐๐โฃ๏ธ

• Is this using the limit definition?

• Gracias NancyPi ๐

• Da

• nyc vediu dear! ummah!

• I need Horezantal Asmyptotes too ๐

• Oh god!! Where will you find a woman like her, who looks so beautiful and is a Mathemagician… She’s awesome!

• At 4:13
Do u just able to predict what the graph might look like and where the curve might be just by looking at the equation? I see u done that a lot.

I would need to find at least 6 different point on graph to draw those. But u seem to just draw it out without calculate any point. Do u pre-calculate it or u actually able to get an idea of what the graph might look like?

• This helped. And thank you for including the steps in the description ๐

• Fuck math can I marry you??

• Tf Iโm still confused

• you tell us what to do but you don`t tell us why

• Thank you! I missed the past week of math class and this really helped!

• It is very helpfull video

• Perfect video, this is what I needed help with. I would have an A+ in your class

• I didn’t get what u were saying, your way too beautiful. I got distracted XD

• aight, okay 10 year old

• omg thank you so much for your videos I genuinely understand what I’m doing know

• you cant use this method, because the cross canceling get’s rid of one of the answers in differently formatted problems.

• Rocket Science! When my professor hears that from me she knows I’m in Trouble. With Nancy explaining it. Well, it does make sense. Again I thank you for taking the time to do these tutorials. I just happen to be in College Majoring in Math!!! and I’m 68 years young. If I can understand it so can You !! Just Don’t give up, but learn to take a break, and get back to it!!

• “…but you can like this video…” – Shrek 2 – puss in boots eyes scene

• Am I the only one noticed she wrote all words reversed??

• Can anyone tell which type of board she is using

• What is the name of this type of writing board

• Ma’am your so beautiful and the way you teach is very lovely and understandable.

• Life saver!

• I WANT THAT TOP

• But Wait !! When she did the factorization and then she canceled out the factors from the top and the bottom, she left with (x+5)/(x+2) which suppose to be a simplified form of the initial one (means any input in the factorized equation should give the same output as the initial equation! ) However, if we put 2 in the simplified equation we get 7/4 but it gives 0 in the first Equation? …. Who can help clean up these shit thoughts in my Mind Plzz thanks

• hold the fuck up she’s writing backwards and that shit looks better than my handwriting forwards.

• This was explained very easy!! I could never graph or understand how to graph lines of vertical asymptotes and solve the functions. This really helped me practice!!!

• Can you make a video about holes how can we find holes

• Very useful and easy method ,thanks mam

• I think this just became my favorite math channel. The ability of writing backwards, her calming tone and voice, her examples and graphs, and the way she explains everything

• Excellent teacher – makes the math easy to comprehend.

• I STAN MY MATH QUEEN!!!