How to Find the Limit at Infinity (NancyPi)

MIT grad shows how to find the limit as x approaches infinity or negative infinity. To skip ahead: 1) For a POLYNOMIAL or CONSTANT in the limit expression, skip to 1:56. 2) For a RATIONAL ("FRACTION") expression in the limit, skip to 8:49. 3) For something of the form (SINX)/X, skip to 23:01. and 4) For an EXPONENTIAL example, skip to 27:27. Nancy formerly of MathBFF explains the steps.
For LIMITS at a FINITE VALUE (not at infinity), jump to the video: • How to Find Any Limit ...
Follow me on Instagram! / mathnancy
Follow me on Twitter: / nancypi
1) For a POLYNOMIAL or CONSTANT in the limit expression: the limit of a CONSTANT (just a finite number like 3), as x approaches infinity or negative infinity, will just be equal to that same constant number. For the limit of a POLYNOMIAL (such as 2x^2 + 2x + 5), as x approaches infinity or negative infinity, just focus on the leading term (highest x power term) in the polynomial, usually the first term. You can ignore all lower terms, because as x gets infinitely large (in either the positive or negative direction), the highest term is growing most quickly, and the lower terms will not affect the limit value. Then figure out whether this leading term will grow toward positive infinity or negative infinity, as x gets extremely large. For instance, if the leading term is 2x^2, as x goes to positive infinity, this leading term will also go toward positive infinity, and the limit will be positive infinity. If the leading term were -2x^2, the x^2 would go toward infinity, as x goes to infinity, but because of the -2, the limit is negative infinity. For X approaching NEGATIVE INFINITY, keep in mind that a negative number, to an even power, becomes positive. A negative number, to an odd power, stays negative. For instance, what if the leading term is 4x^3, and you want to find the limit as x goes to negative infinity? If you think of plugging in a very large negative number for x, the 4x^3 would still be large and negative because of the odd power. The term would go toward negative infinity, so you can write that the limit is equal to negative infinity.
2) For a RATIONAL ("FRACTION") expression in the limit: I show a shortcut (and also the official formal algebraic method) to find the limit, as x goes to infinity or negative infinity. For the SHORTCUT, there are three cases: 1) If the DEGREE OF THE NUMERATOR IS LESS THAN the degree of the denominator, then the limit is equal to zero, no matter if x is approaching positive infinity or negative infinity. 2) If the DEGREE OF THE NUMERATOR IS EQUAL TO the degree of the denominator, then the limit will be equal to the ratio of the coefficients of the leading terms of the numerator an denominator, no matter if x is approaching positive infinity or negative infinity. For instance, if you're finding the limit of the rational expression (2x^2 - 5x)/(8x^2 + 3x), as x tends toward infinity or negative infinity, the limit will be equal to the ratio 2/8, which simplifies to 1/4. The limit equals 1/4. 3) If the DEGREE OF THE NUMERATOR IS GREATER THAN the degree of the denominator, then the limit will be either infinity or negative infinity. For ex., to find the limit, as x approaches infinity, of (3x^2 - 2x)/(x + 5), instead focus on finding the limit of the ratio of leading terms, as x approaches infinity. So instead, you can find the limit of 3x^2/x, which simplifies to the limit of 3x, as x approaches infinity. Since 3x goes toward infinity, as x goes to infinity, the limit is infinity. NOTE: if x had instead been approaching negative infinity, the limit of the original expression would have been negative infinity, since 3x goes to neg. infinity as x tends to neg. infinity.
3) For something of the form (SIN X)/X: there is a trig property you can use to simplify: that the limit, as x approaches infinity or neg. infinity, of (sin x)/x is equal to 0. If your expression isn't exactly (sin x)/x but instead has something like 2x or 3x inside the sine function, like sin(3x) over x, you can use the same property but first have to rearrange the expression in a way that matches what you need. Be careful not to confuse this property with another, very similar, (sin x)/x expression for when x is approaching 0. That property states that the limit of (sin x)/x, as x approaches 0, is equal to 1. Check out my video on how to find the limit, at a finite value, for how to use that property.
4) For an EXPONENTIAL in your limit expression (with a negative power): for instance, if you are finding the limit, as x approaches infinity, of e^(-2x), first rewrite the expression using the reciprocal instead of the negative power, so 1/e^(2x). Then it is easier to see what happens as x gets extremely large and goes toward infinity. The e^(2x) gets extremely large, so 1 over a very large number will head toward zero, and the limit will be equal to 0.
For more of my math videos, check out: nancypi.com

[@ReaperOKnowledge]

Why don't teachers teach like her? My professors teach something simple like "1+1" then give us homework like"If Timmy left home at 9pm, how much does the moon weigh 2.5 million years ago in the future?"

[@Grungebaeby]

Jus dumb shhh

[@RealSugam]

Its cause the vast majority of math teachers have ZERO business teaching anything at all.

[@Achilles21]

Fr fr

[@hercules101jovatronmx7]

I almost peed myself reading your comment....damn that was funny.

[@niina.5190]

So harshly true 😭😭

[@Ralvy]

Do you guys feel that she explains better than DR and teachers in University

[@rashma23]

if she taught I'd take calc online from whatever school she was teaching

[@izzat_kiswani]

Yes 👍

[@asiaatienza6827]

literally revising for exams rn

[@saraelynch]

So much easier to understand

[@wamo693]

Hell yeah

[@bonelesspizza6311]

I'm in grade 12 right now and ever since finding this channel, I'm FINALLY able to breath and study knowing that I understand what the teacher is talking about. I've watched all the videos on the channel (this is my 3rd time watching this vid) and I'm finally confident that I can do this. I have a test on limits tomorrow so wish me luck!!!!!

[@nigateeshetu7087]

me to

[@lionsgate7875]

How did you do?

[@sarujineegogoi929]

Best of luck

[@xXTheDemonK1ngXx]

its been one year how did you do?

[@bonelesspizza6311]

@@xXTheDemonK1ngXx i did amazing, I got an A and I got in u of t

[@yeny7194]

Nancy I dunno how to put into words how much I appreciate your videos. I just started taking Calculus 1, I'm learning about limits and continuity/discontinuity right now and your videos are helping so much.

[@sebastianp994]

Where are you at now after 2 years?👀

[@HackedPC]

@@sebastianp994 3 years

[@ahmedwagdy8432]

@@sebastianp994 where r u after one year😃

[@wonderingswordsman1177]

have u d1ed?

[@idk-xr9wm]

@@ahmedwagdy8432 where are u after 4 months

[@astridj3918]

QUEEN OF MATH

[@SydtheKyd]

+

[@ChalkEater419]

@@SydtheKyd -

[@eunhabunny2382]

Yes

[@edgarvicente8859]

For me she is my Queen of Heart...

[@robertowisconetti2732]

@@edgarvicente8859 siiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiimmmmmmmmmmmmmmmmmmmmmmppppppppppppppppppppppppppppppppppppppppppppppppp

[@camd3673]

Nancy, if you are reading this please just know I am so grateful for your videos... so many difficult concepts explained perfectly... Thank you so much!!!

[@ferreiraronald7873]

Nancy explains at just the right level, she explains the logic and the practical procedure, without always needing to explain how to derive the fundamentals. For me this is ideal, as my 14 year old daughter in 9th grade now needs help and I have forgotten a lot of the basic stuff in math from university days. Listening just once to a video from Nancy on any topic is enough for me to help my daughter. Of course Nancy explains a lot better than I possibly can and my daughter would be better off just listening to Nancy's videos directly but for now she needs the interactive mode to ask questions. In the future she will be better off listening directly to Nancy's videos. Sure Nancy explains better than most teachers (certainly my daughter's math teacher). But many teachers both at school and university are sadists and get their satisfaction mystifying students. Thank you Nancy.

[@chriswilliams5291]

Today is my first day studying calculus. I just finished reviewing your limit videos I'm feeling pretty confident about limits. Thank you for the excellent explanations on how to solve limits! :)

[@poochizzle]

I usually never comment on youtube videos but I just had to say how much this has helped me. THANK YOU SO MUCH and never stop making math videos!! :)

[@mons9974]

To skip ahead: 1) For a POLYNOMIAL or CONSTANT in the limit expression, skip to 1:56. 2) For a RATIONAL ("FRACTION") expression in the limit, skip to 8:49. 3) For something of the form (SINX)/X, skip to 23:01. and 4) For an EXPONENTIAL example, skip to 27:27.

[@crazyxkid212]

She explained how to find derivatives exactly as my Calculus teacher explained today I wonder if he saw your video before class.

[@asmaesabri5857]

Wow, you are so lucky to have a teacher explain like this My teacher all the time just writes on the blackboard and then wipes

[@rebelplayz5639]

And am studying that shit at grade 12th.

[@kqnrqdtqqtttel1778]

Same with me, I am studying this in 6th grade

[@rebelplayz5639]

Well guess what im not gonna need to study this shit anymore. Exams cancelled. Promotion granted😀😀😀

[@platinumdilan1726]

Honestly we need more teachers like this in the world

[@simphiweharvey2258]

Nancy>>>Keep helping us.God will bless you

[@noscamplease6152]

Thanks from my side.

[@bensmyth450]

Nancy, you are a godsend. I'm taking Analysis of Algorithms and have completely forgotten how to evaluate limits until now. Thank you!

[@jacquelinealmazan9169]

PLEASE do continuity

[@usamarafieck1715]

Do some more limits.

[@matthewanderson1916]

oh my god your videos are so helpful, probably one of the best, if not the best youtube instructors i've found for calc/precalc in my many hours of searching the web. so clear and simple, explained in simple terms, good examples, well lit and neat handwriting, organized well, clear speaking, the whole package. thank you so much.

[@jeremyankamah4737]

You have really cleared my confusion.. please never stop making these videos...you are helping the confused souls in class make it!!!!

[@kalidas996]

She looks so confident and look how focus she is while writing ....that everything seems so easy...

[@diana6739]

Hello, Nancy! I'm in grade 12 and taking Calculus I, and I'm so thankful for your videos. You make it so much simpler to understand. Thank you so much, Nancy!

[@duncamacintire7591]

Fantastic!!! She's cleared up a MASSIVE confusion I've been harboring since my first semester at engineering school

[@joellabrador8937]

I’m amazed at how she’s able to write in reverse.

[@elisiajimenez5527]

I think she just mirrors the video lol

[@mustafarasul1346]

bruh you really think she writes it in reverse lol

[@estefaniareis6637]

@@elisiajimenez5527 could you explain how to do it

[@alfedrale6660]

@@mustafarasul1346 r/woosh

[@josephzimba9348]

Nancy you explain very well and you take your time. This is good staff. Your smile makes the difficult mathematical calculations simple.

[@daviddial6147]

I love you Nancy! You are helping me through engineering one class at a time.

[@joane7651]

I was confused on other tutorials. Because of your shortcuts, everything makes sense now. Thanks!

[@kdmjr0225]

This made limits a lot easier to understand. Appreciate it.

[@nadineamr9372]

You're a blessing!! I'm in uni and we are going by topics each week like a breeze and I'm always so lost until I watch your videos. Thank you!!

[@astridj3918]

WE LOVE A LEGEND

[@Watchermelon0725]

I mesmerized by your beauty ✨ it feels like an angel is teaching me about calculus

[@hagaracherrat7919]

Keep going, I am in love with your videos. You make it seem so simple and explain it in the most basic way for everyone to understand. Thank you so much.

[@nicholaswilliams5996]

You explain limit better than anyone else! You've made this so simple to understand for me thank you for making these videos!!

[@mohamedmire8646]

This is making Maths very intersting. When ever it comes to searching a Math help I remember this channel. Thanks for the amazing explanation Nancy

[@adityadaila4669]

Thank you mam.....I am from India And I haven't seen a teacher like you very experienced in explaining and interacting with the students.... Hatts off mamm....🙏

[@kellyjia130]

I really appreciate the great structure you design to elaborate on this! it makes things more clear for me to understand.

[@thabolikotsi5765]

Girl what did you get me into 2017 I watched one your videos fell in love with now 2021 I'm a math and applied math major doing graduate math courses wish me luck😁

[@chicharronejr5847]

Came here to get help for my calculus test today. Instead I just fell in love

[@leovolont]

Yeah, it's good to take a Cold Shower first so you can pay attention to what she is saying.

[@robman2095]

AND she could do your math homework for you!

[@Coreyahno]

ChiCharrone Jr Cringe

[@victoranaya7548]

I learned calculus and fell in love!!

[@mestruorlan1763]

I got the same feeling every time I watch some of her math tutorial videos.

[@powerbuff2054]

srsly you need more subscribers. you've been helping me so much through college!

[@Shinwaar]

Thank you so much for helping me in math but I would appreciate it more if you drop your skin routine 💕💕

[@taetaestata7685]

I'm a senior in high school and i'm really grateful for u Nancy:) thnx Ur student from Algeria!!

[@anaaya3131]

This was very well explained! Who else comes to her videos straightaway when you don't understand a topic in the class? *.*

[@leovolont]

Hi Aymoreno, you asked six months ago " Who else comes to her videos straightaway when you don't understand a topic in the class? " Well, I watch Nancy Pi's Calculus Videos BEFORE going to class so that I will already know WTF is going on and so my eyes won't glaze over.

[@CP-28]

I would come to her videos even IF I understood the topic in class

[@user-nu2if9kn9j]

Thank you so much. Your videos are SO MUCH more understandable than any other resources including my teacher 😭 This helped so much!!

[@aishahcarana-masiroh6691]

Thank you so much, you are just amazing - I'm always blown away by how incredibly simple you present these topics.

[@bernardolarios9996]

thank you Nancy I can watch your videos all day :)

[@cmejia4401]

Thanks again Nancy! I always like how you simplify everything.

[@tshiamomadikgetla1716]

Literally just understood 2 weeks worth of work in an hour and 30mins...Protect this woman at all costs guys😭😭

[@aljoparedes8667]

Finally found a video that relaxes you and gives you a headache at the same time :D

[@goldmist_1216]

I am feeling like crying!!! Why I find her today not before!!! OMG!!! Thank You Sooooo Much Dear

[@jacquelinealmazan9169]

You're amazing! I wish my calculus professor taught like this.

[@hhmmhmhm5743]

You did a better job of explaining this in 30 minutes than my professor did in 80 and my textbook did in 20 pages. This truly highlights how vital proper teaching methods are for the students' comprehension of the material. Thank you!

[@Bewoulf1223]

Nancy is back! Yay!

[@childrenofkoris]

thank you, you are a life saver, i dumped math since i took calculus when i was in my bachelor courses, now i am a full time employee for a firm, the only reason why i still use advanced math is because i am doing programming and development

[@mogli3leo]

Thanks for this video! In 30 Minutes I understood what 6 Hours of Classes couldn't achieve..

[@fanwellnyereyemhuka8378]

This is excellent Nancy.You inspire me a lot.I am doing Real Analysis as a second year course in Bsc Mathematics.Keep up the marvelous work.

[@qupy3391]

thanks to your help, i can slide through my hw. saving myself hours of agony.

[@asmaesabri5857]

I think just me and you who are new to see this video lol

[@AV4Life]

Something about you just makes me learn math! Thank you

[@henrytjernlund]

I wish these kinds of videos existed back when I was in college (1980, give or take.) I would have done much better.

[@leovolont]

Yeah, but back then it was a War about the Best Textbooks. but, yeah, Information was different in those days. To KNOW anything back then, you had to subscribe to some Specialty Magazine (Mathematics Magazine) or join a club. But, yeah, you just couldn't Google anything back then. You knew or or you didn't... researching anything was like climbing Mount Everest without a sherpa.

[@sanjaikmrai12]

I was trying to recall the topic to help my son whose school is closed due corona pandemic and I went through all her limit vedios in one go . Brilliant .

[@terminatr]

Thank you sooo much for this!!! My teacher is terrible and has no idea how to explain things to us. I always end up having to go to someone like you to figure it out! 😂

[@lylygior8332]

Thank you for making this. I have to retake a test on this tomorrow, well i guess today now, and have been trying to cram all night desperately trying to figure this out until I came across your video. I truly can't thank you enough.

[@keekli9046]

I love how u explain things!

[@HOLYLIFEIFY]

What comes to mind is to earth come angle my ear gates and eye gates receiving at the words "lift up the sword that's next to my throne you are sitting on Terrill" and so then lifting up the real live sword this being when I was in 2nd grade, was a negative infinity also a positive infinity. Terrill TC!

[@samkelenibe2018]

Nancy. Euler's great granddaughter. 😀

[@leovolont]

Yeah, to Euler little Nancy was always his number e.

[@AlexandrBorschchev]

Tf

[@samkelenibe2018]

@@leovolont 😂

[@miktheo328]

Best, calm, cool, collected, methodical explanation ever! @HorribleTeachers, please use this video as a self-improvement tool.

[@scottcarter2981]

plot twist: she doesn't fast forward her drawing/writing, she is actually a robot or superhuman and just does it at super speed

[@Thabang_Kgatla1]

Thank you Nancy, I am Thabang from South Africa

[@Sometimes_Always]

She's so ridiculously pretty..

[@ngathawilliam1760]

🤍i completed mechanical engineering (diesel) from college and industrial engineering higher certificate at university for maths, your videos helped me a lot nancy 🤍 much appreciating💥❤

[@johnlloydmichaelpangan4431]

Why do I get the vibe of Dakota Johnson from Nancy because of how she talks in a calm and chill manner. Lol

[@princesspat5239]

You're great at explaning but can you also explain what you do to your skin? Because it's glowing so much

[@joemonem]

not gonna lie, its hard learning when the teacher's hot 🔥

[@JuanCruz-et8so]

I can’t even focus, she makes sense but she keeps trowing off my concentration every time she smiles. ♾ smile 😊

[@MrMrREmm]

So for rational expressions limits to infinity are the horizontal asymptotes?! 🤯🤯🤯

[@leovolont]

Hi MrREmm, You say "So for rational expressions limits to infinity are the horizontal asymptotes?! " Well, almost. If the Denom is a higher power than the Numerator then the horizontal asymptotes will be at Y = 0. If the Num and Denom powers are the same, then the Horz Asymptotes will be at the Fraction described by the leading coefficients. But if the Rational Expression is TOP Heavy, that is, if the Num out powers the Denom then you WON'T get a Horizontal Asymptote. There you have to Simplify your Leading Terms of both the Num and Denom and see what kind of Waveform that gives you (if the difference between the Powers is 1, you will get a Straight Line with a Slope, if the difference is 2, you will get a Parabola, etc.)

[@kylemeyer4919]

She always gets to the point it helps me a lot when I am trying to do calc homework

[@AlinaKay47]

I missed you math queen

[@tarcisiomenezes3318]

thank you so much, Nancy. You are doing an amazing job

[@jesypop2316]

I'll be thanking you during my speech at my grad party

[@anaisnin2433]

I literally love you so much you're changing my whole calculus experience! ugh sooooo grateful for you!! THANK YOU!!!

[@qq_mingle]

Hi @NancyPi thanks for ur outstanding videos. Can you explain Continuity of functions under the topic Limits n Continuity. Thanks in Advance 😊

[@claytonjohnson3133]

Another way of thinking about how sinx/x functions are always equal to zero (as x approaches infinity) is by realizing that the sin of any number, no matter how large, always hovers between -1 and 1. If you divide a number between -1 and 1 by an infinitely huge number you get essentially zero.

[@biyopist]

Functions or expressions may have a limit at infinity but Nancy has no limit in mathematics :)

[@PhamNguyenLinhLan]

Nancy thank you so much! You explained this better than any teacher I know.

[@erickmartinez5513]

welcome back

[@circuitwave]

You're amazing! This video really assisted my understanding of Limits at Infinity!

[@markjhoncaliboso2557]

My GOD I can't focus I always get destructed by her smile

[@titanian2141]

Like bro ..lmaoo

[@aFreeman0409]

I would have never know that infinity had a limit without you Nancy...

[@kylefrank638]

*realizing I'm understanding this* "Hey, cool!" *realizing one third of the way through the video that she's having to calculate all of this backwards on the board* ._.

[@TeddySlump]

I'm pretty sure she writes normally then just flips the video in a video editor

[@velocreators2250]

This videos are amazing I just got my 3 hours of class content in a single video thank you!!

[@mellorine4300]

Really curious how she writes in reverse, or is she? someone explain pls

[@VeritasEtAequitas]

Right handed, normally, on glass facing a camera on the other side. Then just flip left/right like a lot of webcams do anyway. The fade-out effect of just the text is a little more complicated but I figure out at least one way she might be using.

[@pulkitraj1189]

hey Nancy it's a very great incidence that i have find out your channel after having so much wandering on KZbin in order to get perfect learning as you provide ;LOVE FROM INDIA

[@saqibshams5143]

Hey!! I noticed a thing. Her facial expressions are a lot similar to Dina fitz titan. From attack on titan. 😶😐😐😐😶😶

[@ronycb7168]

Thanks will be sure to watch this whenever I need to get a quick memory refresh

[@yazn2322]

Welcome back

[@DeepakTiwari-vj9fi]

If teacher like you is my professor i swear i will never miss my mathematics class, yo are so gorgeous and the way you explain is simple and awesome.

[@angelolimbaring9681]

It really helps a lot when the who teaches is cute.

[@aasrabaig2748]

Nancy is my ideal math teacher.

[@motaungthatosetjhaba9427]

you are beautiful also your math is beautiful , the up the good work

[@vibe.2257]

Thank you Nancy for the videos about limits. I understand it better with you. God bless and more power!

[@TridentEditing]

You're beautiful

[@lucybusselle2182]

THANK YOU SO MUCH! My calc prof sucks and these videos have helped me so so much

[@BrodashMx]

Nancy yo are so beautiful i will follow you to the Infinity. ♥♥♥

[@Sleepyhead64733]

Geekin

[@VarijSaini]

simp

[@kamalkaushik8053]

🤣🤣🤣🤣

[@EchoOfAmarok]

I am a Sophomore in college (about to be a Junior) and still finding this very helpful lol. Thanks for the videos!