Introduction to Limits (NancyPi)

Published on August 31, 2018 by

MIT grad shows what a limit is, how to read the notation, what it means on a graph and how to find the limit on a graph. To skip ahead: 1) For how to understand limit NOTATION and the CONCEPT of the limit, skip to time 0:34. 2) For WHICH WAY TO LOOK AT THE GRAPH to find the limit, including when to use the X and when to use the Y, skip to time 1:52. 3) For ONE-SIDED LIMITS notation, including the LEFT-SIDED LIMIT and RIGHT-SIDED LIMIT, skip to time 7:54. 4) For how to understand limits where X APPROACHES INFINITY or negative infinity, skip to time 10:24. Nancy formerly of MathBFF explains the steps.

For HOW TO FIND THE LIMIT (at a finite value), jump to

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1) LIMIT NOTATION and WHAT A LIMIT MEANS: You can read the limit notation as "the limit, as x approaches 1, of f(x)". This means "when x gets very close to 1, what number is y getting very close to?" The limit is always equal to a y-value. It is a way of predicting what y-value we would expect to have, if we tend toward a specific x-value. Why do we need the limit? One reason is that there are sometimes "blindspots" such as gaps (holes) in a function in which we cannot see what the function is doing exactly at a point, but we can see what it is doing as we head toward that point.

2) HOW TO LOOK AT THE GRAPH to find the limit: a) For a removable discontinuity (hole), b) For a removable discontinuity with a point defined above, and c) For a normal line. When you're finding an overall limit, the hidden, implied meaning is that YOU MUST CHECK BOTH SIDES OF THE X-VALUE, from the left and from the right. If both sides give you the same limit value, then that value is your overall limit. In our example, to find the limit from the left side, TRACE X VALUES from the left of 1 but headed toward 1 (the actual motion is to the right), and check to SEE WHAT Y-VALUE the function is tending toward. That y-value is the left-hand limit. To find the limit from the right side, trace x values from the right of 1 but headed toward 1 (the actual motion is to the left), and again check to see what Y-VALUE the function is heading toward. That y-value is the right-hand limit. Since the left limit (2) and the right limit (2) are the same in our example, the overall limit answer is 2. If they were not the same, we could not give a limit value (see #3). IMPORTANT TAKEAWAY: For the limit, we DO NOT CARE what is happening EXACTLY AT THE X-VALUE and ONLY CARE what y-values the function is hitting NEAR the x-value, as we get closer and closer to that x. In other words, the limit, as x approaches 1, of f(x) can equal 2, even if (1) = 3 or some other number different from 2, or even if f(1) is not defined or indeterminate.

3) ONE-SIDED LIMITS (RIGHT-SIDED LIMIT and LEFT-SIDED LIMIT) for a jump discontinuity: as you saw in #2, to find the overall limit, you have to check both the left and right limits. Sometimes the left limit and right limit are not the same. If you get a limit question with notation in which the x is approaching a number but with a plus sign or minus sign as a superscript, that is notation for a one-sided limit. The minus sign means the limit from the left, and the plus sign means the limit from the right. IF THE LEFT limit AND RIGHT limit are NOT THE SAME, then the overall limit DOES NOT EXIST (sometimes written as "DNE"). Even if the left and right limits are different, you can still write the left-sided limit and right-sided limit values separately.

4) LIMITS in which X APPROACHES INFINITY (or negative infinity): Another "blindspot" is when x goes toward infinity or negative infinity. Since we can never "see" exactly at infinity (or negative infinity), we can use the idea of the limit to say what y-value it looks like the function is headed toward when our x value approaches infinity. If x is approaching INFINITY, TRACE x values TOWARD THE RIGHT (the large positive direction) on the graph, and see what y-value the function is approaching. That y-value is the limit. Note that the function may be approaching an asymptote. If x is approaching NEGATIVE INFINITY, trace x values TOWARD THE LEFT (the large negative direction), and check what y-value the function is getting closer and closer to on the graph. That y-value is the limit.

For more of my calculus and precalculus math videos, check out:

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  • عبدالله 3 years ago

    كيف تكتبين وما يطلع بالعكس

  • DozyProductions 3 years ago

    Crazy, what she says is as good as how she looks. Thanks for giving a great explanation for us.

  • MS 3 years ago


  • M-Ahmad Nasir 3 years ago

    im not here to learn 🙂

  • Kevin 3 years ago

    Can you be my PRIVATE tutor?

  • sirjohnnyofAwesomness 3 years ago

    Is she writing backwards!? Lol wtf

  • Marek Krakovský 3 years ago

    Wow, a beautiful mathematician!

  • Saygın Tan 3 years ago

    if ı have got like you mat teacher ı am be professor

  • Rajendra Khedekar 3 years ago

    I really liked the description below the video. You explained the whole thing there also.

  • prum chhangsreng 3 years ago

    Okay, having a difficult time focusing but that help me understand it a bit more for how it using the word Approach instead of Equal. Just like how Y could never reach asymptote, but as X goes on, it gonna get really close to the asymptote. And that also make sense for why we dont care about the exact point, but the point around it.

  • XENDIOUS ! 3 years ago

    Damn! I am just focusing on her.

  • Aigul Zhaparova 3 years ago

    Thank you a lot!

  • Adam G 3 years ago

    Not to confuse things, but….the third example made sense EXCEPT for the fact that functions CAN cross horizontal and oblique asymptotes. Only vertical asymptotes are domain restrictions (cannon be crossed since the function is undefined at that domain value). Horizontal and oblique asymptotes are more concerned with the end behavior of the functions as they approach infinitely (both positive and negative)…SO, how can you determine the limit at infinity when you can see out that far so you can’t know for certain that the function will never cross it? Yes, it approaches 0 and -1 near (x=0), but how can we make that assumption? (We CAN make that assumption with vertical limits because the function will always approach but never cross it as I stated earlier). Just curious! Great video and thanks for the fundamental explanations before jumping right into solving problems. Much appreciated and I look forward to watching more of your videos!

  • Ali Eroğlu 3 years ago

    Aq şuna Türkçe altyazı ekleyin . Kızı kesmek için geliyorum bari bir bok öğreneyim .

  • shoaib ahmed 3 years ago

    hi dear i have a confusion

  • ingeri japhet 3 years ago

    thank you so much this was so much helpful

  • Jackey Tsui 3 years ago

    im sorry teacher i just can’t focus

  • Cao Moleti MOLOI 3 years ago

    You sure are a lifesaver, Nancy. It is through you that my journey in math is SMOOTH. Even my two math teachers combined cannot surpass your tutoring skills. Keep up the tremendous work!!!!

  • david coronado 3 years ago

    Oh no Nancy I can’t concentrate I keep staring at you.

  • FUTURE 3 years ago

    my calc teacher this week spoke too fast in a thick accent while teaching us limits so I couldn’t understand or even pay attention, so after class I find this video and now I’m too distracted by how attractive Nancy is… I’m fucked

  • Aaron Stewart 3 years ago

    I appreciate your explanations, It’s been a while since Calculus I for me, and for tutoring, I find your videos to be extremely helpful.

  • Alex Vong 3 years ago

    not to be rude, but i dont know where to look

  • Brandon Pearce 3 years ago

    Thank you Nancy. You got me through Calc 1

  • Nurdin Kuehnel 3 years ago

    Bend them. And move across.

  • السعيدعبدالغنى السعيد 3 years ago

    I love you

  • Laercio Civali 3 years ago

    Please integrate 0 to inf {1/[e^x sqrt(sinh2x)]} dx. Tks!

  • Paul Price 3 years ago

    I’d like to take you out for dinner to discuss more on limits. Check yes or no.

  • Panda Shiro 3 years ago

    OMG! Thank you so much, Nancy! I really liked your video! It helped me a lot. 🙂

  • أستاذ مرتضى صبيح 3 years ago

    شكلج مال نوم وزواج حرام تخربي بالرياضيات

  • Tom Scott 3 years ago

    so, your explanation of limits/equal signs right around 5:00 kinda backs up an argument i had on another thread. the limit of something is _not_ really equal to that thing, but is what that thing _approaches_ . without referencing euler’s notes, you back me up quite nicely, thanks!

  • Eric Alejandro 3 years ago

    the limit is blowing my head with a gun

  • mccohn47 3 years ago

    You are literally a GOD SEND TY <3

  • Anna Grace 3 years ago

    absolutely gorgeous & SO appreciative for your explanations. you simplify complex ideas better than any professor 😭♥️ thank you so much

  • lachikagreen 3 years ago

    I finished my homework in 10 minutes thanks to your limits videos. Please post more! 🤓

  • birddogstudio 3 years ago

    Thank you Nancy. These videos are incredibly helpful.

  • MeMyself & I 3 years ago

    It’s honestly very hard to learn limits from such a beauty. Luckily for me, I did these way back in my college many years ago☺👋

  • Abdulmalik Umar 3 years ago

    Very beautiful

  • Yapi Bowler 3 years ago

    For me, you have the perfect lecture videos in YouTube ma’am! The discussion is highly commendable. The speed, the presentation, and all, are just so perfect to me. I hope you make lecture videos to all topics in calculus. I am really amazed with how you make calculus looks easy. Thank You!

  • Humanbeast S 3 years ago


  • William Fairbanks 3 years ago

    Nancy, you have no limits. lol Thanks for teaching math. These poor dudes though. 😉

  • Trevon Friday 3 years ago

    I am self-studying calculus, and I have been trying to learn limits for a whole day. Your video really helped me understand limits. Thank you!!

  • 썸원콜더닥터 3 years ago


  • Fahd Ciwan 3 years ago

    i had to close my browser when my mum came in the room …. i mean, i was doing nothing wrong, just learning maths but u never know with these moms

    • Aditya Ajay 2 years ago

      from which country are you bro .

    • Hardek A. 2 years ago

      @Aditya Ajay India

    • Aditya Ajay 2 years ago

      I got a feel with your language , same here.
      mummy boli, “ye kya hai , maine kaha calculus ,
      mom had some doubts “.

    • The xD 2 years ago

      @Aditya Ajay exactly I’m also from India and got the same name lol and got the same reply from my mummy too lol

    • The xD 2 years ago

      @Aditya Ajay nah im in 10 currently preparing for ioi

  • Domenico Grimaldi 3 years ago

    6:49 …how did you simultaneously draw that line and erase the circle?

  • Peter Cooper 3 years ago

    this content is awesome!!! very helpful

  • Ranu Thakur 3 years ago

    So sexy my dear Nancy ma’m!

  • Aniket Jhariya 3 years ago

    Hey, Welcome back everyone. My names is NaaancyyPiiii !!

  • S tiwari Tiwari 3 years ago


  • bilal 3 years ago


  • Nicholas Tracy 3 years ago

    WHAATTTT?? Still searching youtube for math help?
    Are the videos from 2006 realllly helping?

  • amresh yadav 3 years ago

    she is explaining limit, without defining an arbitrary function.

  • Programming Interview Prep 3 years ago

    How do we know the limit is 2, and not say 2.00000001?

  • Blue Star Industrial Arts 3 years ago

    Would she make these vijeos if she wasn’t so hot?

  • Román Fernando Ruch Román 3 years ago

    Excellent explanations, congratulations

  • Garry Singh 3 years ago

    Great man g

  • B P 3 years ago


  • Arati Mohite 3 years ago

    Good one

  • ethan Millan 3 years ago

    thanks for the video it’s a great introduction and helped alot

  • Sparky McFarlane 3 years ago

    Where was a math teacher like this for me when I was 13?

  • Alexandra Kaidan 3 years ago

    Limits? I come from a family of computer engineers, Oracle Clinical, Lockheed, NASA, are a few of the places my mom has worked! She holds several degrees! Tell me, what do you do? Because this isn’t your profession.

    What do you do?

  • Mohamad Harith 3 years ago

    couldnt focus on the lecture when the lecturer is extremely attractive. i rather watch sal khan from khanacademy videos.

  • John Barlow 3 years ago

    Cute,smart, and able to explain,(teach)! Wow the whole package!,

  • John Barlow 3 years ago

    Have you ever thought about allowing schools to make use of your videos? I have had 22 years of school,and your videos are the best explanation of basic math I have ever seen!!

  • John Barlow 3 years ago

    Have you ever thought about allowing schools to make use of your videos for teaching? I have been in schools for more than 22 years. Your videos are the best explanation of basic math concepts I have ever seen.

  • Herramientas Académicas 3 years ago

    Hi, Nancy I like your videos, Congratullation !!! I am starting my Chanel in Colombia and I would like to know what is your technique for writing…Thank you…

  • Rahmat Abadi 3 years ago

    You were using the learning glass that was invented by a physicist (I forgot his name).

  • Hinks Ben 3 years ago


  • Chris Greek 3 years ago

    Nice …parabolas!

  • Maxim Balakin 3 years ago

    Thank you so much for doing my teacher’s job for free

  • esteban fraile 3 years ago

    Thanks from Argentina in the very south of the world…

  • The Teen Engineer 3 years ago

    i wonder how i can use this in my computer programming

  • Zulfiqar Ali 3 years ago

    If you were there when I was in college today I would have been a professor

  • some person 3 years ago

    Omg thank you so much but you dead sound like a gun is being held to you while you say all of this 💀 💀 no hate just a funny thought

  • Jiaru Liu 3 years ago

    I just don’t understand why are there people who doesn’t like her video, come on ! she is sooooo pretty and freaking smart!

  • RedRedMCmusic 3 years ago

    if i had you as a teacher in high school i would of been better in math. instead my math teacher looked like a cow and was a bitch.

  • kun mcqueen 3 years ago

    nacy chan saiko

  • James Lindgaard 3 years ago

    Would it be just as easy to say that when f(x) = 2 that the limit defines <2 and >2 which would be the actual definition of the limit which is why itself is undefined? This would be because it shows a transition. A basic example might be f(x) = 9.8 m/s/s. It would show f = ma between an (x) value of t in seconds. It would give force 2 ranges in the domain of seconds. it’s like if the speed limit is 45. Below 45 is safe, above 45 is not safe, can 45 be both safe and unsafe? 🙂

  • Moon River Valley 3 years ago

    oh i love math !!!

  • Made Sudiasa 3 years ago

    Hello Nancy, I’m Made from Indonesia. Nice to see your video.

  • realeboga monei 3 years ago

    I salute you my dear sister

  • Sunil Agarwal 3 years ago

    You are looking nice

  • Mohan Joshi 3 years ago

    You are talented, and very beautiful at the same time.

  • Suryapraharsh MATSA 3 years ago

    mam please explain the definition of limit

  • Shazzy Kitawa 3 years ago

    never been able to understand youtube math videos, THIS HAS MADE EVERYTHING EAASY

  • BATTL EX 3 years ago

    Best video on YouTube

  • Dum Dum 3 years ago

    I’m so very grateful. All I can do to repay you is to watch every commercials. No skipping.

  • velveteencherrytart 3 years ago

    these comments are nasty and 90% from old men…why are y’all watching high school math videos lmfao

  • Rick Eastburn 3 years ago

    How can you/they say the function (Y) does not equal 2 at X =1? It is quite obvious that at X variable 1, the function (Y) equals 2.🤨

  • Sayuj P 2 years ago

    Ohh i seriously wanna learn this but your face, ohhh god dammit!!! makes me distracted….please don’t show your face…but i did managed to watch the full lecture so don’t judge me 😚

  • Ponce Alvin Legaspi 2 years ago

    So Smart and Cute Teacher.

  • MATH WORLD 2 years ago

    How you write? I need like program

  • Gaming Wizard 2 years ago

    The only matheatics videos from which i dont get bored!

  • abhineet kumar 2 years ago

    how do you write like that?

  • Michael Giler 2 years ago

    If you were just this incredibly beautiful you would still be a goddess. But your mind makes you the goddess of goddesses. That you are sharing your beauty and knowledge with the world makes you the goddess of goddesses of goddesses.

  • bijay kumar mondal 2 years ago

    You are very beautiful

  • shakila sanjeewa 2 years ago

    You are a great teacher! When my professor was teaching me these things i got mad..😒 i thought she is teaching us by using another language bcs i didnt understand anything !

  • Kotha Lakshmi Kusuma 2 years ago

    ur awesome … just great … gave me interest to study math

  • Sunny Rana 2 years ago

    Mam u r very cute…….

  • thanuja suraweera 2 years ago

    Thank you so much for all the videos on calculus Math. It really helped me a lot when I was taking math 114 at university.

  • Adriano de Morais 2 years ago

    Wooww! So pretty! 💕💕💕💕

Comments are closed.