# How to Find the Limit at Infinity (NancyPi)

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Published on May 15, 2018 by

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: https://youtu.be/nJZm-zp639s

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: http://nancypi.com

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• welcome back

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• Shouldnt it be e ^1/infinity which tends to e^0?

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• 0/0 is indeterminate means that limit doesn’t exist or you didn’t simplify enough. Infinity/Infinity is simply 1 and -infinity+infinity should be 0 I believe but you also have to make sure that you account for the coefficients in front of x

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• 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.

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I wonder if he saw your video before class.

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My teacher all the time just writes on the blackboard and then wipes

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salute to you……

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• 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.

• Did she learn to read and write backwards just for the sake of making convenient videos for us?

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• 0(∞) is indeterminate

• Hi Drin, you say “0(∞) is indeterminate”. Really? Do you have a proof for that. My intuition is that if you have A(B) and A is Zero and B is any magnitude, then the Product will be Zero. And Infinity is simply a Magnitude, that, yes, in indeterminate in itself, BUT we are really CERTAIN about Zero, aren’t we? It is a case where once we know that that A is zero, the Product is already Determined and we can ignore the Value of B. Doesn’t that sound right to you?

• @Leo Volont We know that 1/0=infinity so 0*(1/0) = 0/0 (0/0 – indeterminate)

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

• 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.)

• Amazingly clear and easy to understand!

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• 24:00 how did it disappear?

• Ah yes the magic of editing

• Thank u very much ..

Miss Nancy
Love from manipur

• I wish ı had math teacher like you then ı would be a professor🤥Come to Turkey,Come to Adana.

• How can you write?

• Hi Piseth, you asked “How can you write?”. Well, we discussed it another one of her videos and decided that she write from behind the glass but then uses the Video Editing Software to flip it to Mirror Image. That would sound about right. Oh, when doing Selfie Videos, always check to see whether you look better natural or in mirror image… there is a difference. For instance, all of Tom Cruise’s movies are shot in mirror image because he thinks he naturally looks short and ugly.

• Thank you Nancy!

• A tangente betwen 2 hyperbole plz

• How did i fit thé camera as Well as mirror plz

• You can do the Mirror Image with just the Photo Video Editing Software. A lot of people use it because they think their facial symmetry looks better in mirror image.

• Hi Nancy, I learned something today about your name. I was putting off studying Math (its summer break and next semester is Calculus I, but I’m using you to get a grip on both Calculus I and Calculus II). But, yeah, I was reading one of the most famous Novels in the English Language: “Clarissa” by Samuel Richardson, a novel written in the form of Letters going back and forth between the main characters. Well, Clarissa’s (Clary’s) best friend is Anna Howe, but whenever Anna gets particularly ‘saucy’ the name was becoming ‘Nancy’ which was a kind of a WTF moment for me because, you know, where did this Nancy just come from. So I looked it up and it turns out that Nancy was the diminutive form of Anna and only started to be used as a proper name in the 18th Century… so the change was happening even while this fictional Anna Howe was both Anna, formally, and Nancy, when she was being ‘saucy’. Cool, huh?

• Intresting topic and good explanation. Thank you!

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• Kinda hard to focus… aghh.. I’m kinda attracted to gorgeous intelligent instructors.. I swear I won’t fail my exam, Madam =~=

• She smiles after she writes down on the glass panel

• how to find log limit values?

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

• 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.

• @Veritas et Aequitas how does she have a piece of glass at home

• Thank you so much for your help! I love you as much a Khan and more than Richard Rusczyk!
.

• Every time you are showing but not showing…

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• How will i be determined if the function grows “Faster Than”, “Slower Than”, or “Equal Too” ?

• than another function? Well if the two functions are equal then they grow at the same rate. For faster or slower it’ll depend on what x is approaching or what the function is, I think.

• YES THANK YOU IM IN CALC AB AND I REALLY NEEDED THIS LOL

• 17:50 When I use a graphing calculator to graph the function (3x^2 – 2x) / (x + 5) a vertical asymptote prevents the graph from approaching -5 so the limit of that function as x approach -inf must be -5?

• that -5 is not included because if you use it, you would be dividing by 0, and you can´t do that. The answer to a limit represents a Y value not a X value.

• i don’t feel like im learning anything new… i know all this already but im still watching 🙁
i want you to be my mommy T_T

• Can’t focus hahahaha

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• Hey i m from india i know a same teacher like her as pretty as intelligent.
Her channel is” neha aggarwal mathematically inclined.”
Hope u guys will get something from her also

• Neha aggarwal
She is from Vedantu team right

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• I think she flips image so it looks normal to us.

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• thank you so much this video so helpful ,but i have confused about infinity times zero how is equal zero ,and we know this quantity is undefined

• I don’t know if this will help, but when I first learned about multiplication, I was taught as a number (a) and however many groups of that number (b). So 5 times 4 would be 5 groups of 4 (or 4 groups of 5). I think of it the same way with infinity and 0. You either have unlimited groups of 0, which is still going to equal 0, since there is never any value. Or, you could have 0 groups of infinity, in which 0 groups of something just equates to 0. Hopefully this helps!

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

• *so did she write it forwards or backwards…*

see this why I can never pay attention in class

• She wrote it normally the camera just flips it because the camera is like a mirror

• i love u

• Hey! Thank you for a very helpful video. I was just wondering how the limit of x*[(sinx^2)/x^2] as x approaches infinity would be 0. The expression simplifies to infinity * 0, which is undefined. So would the limit not exist?

• Anything times zero will become zero. Approaching infinity is like a huge number, and a huge number times zero is zero. The expression might simplify to infinity * 0 but the denominator is still 1 so it isn’t undefined, it’s just zero over one, which is zero.

• 18:25 “use algebra! *hopes the idiots she is talking to know what that is*” :D. These videos seriously helped me. Thank you and keep it up.

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• your explanation is very clear.

• Related rates? That would be super helpful!

• Finally I concentrate in study

• Great

• Great video btw, it appears as if you’ve taken an infinite amount of antioxidants, exponentially and nonstop.

• My business calculus ass gets so confused when these sins and Es come out.

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

• Thanks a lot mam .you made it easy to understand the limits for me. I waiting for more videos from
You.I expecting video on complex number

• If she taught these concepts this slow the class would be behind. I mean this is a good explanation but University professors can’t take this long in explaining a concept

• Hloo I like that video

• Can actually use L’Hospital’s Rule for finding the infinity limits.

• Thank you so much… your videos are so clear and you nail what a lot of people would like to find answers for in such a godlike way…

• Are you writing this all backwards? That’s pretty impressive

• Oh yeah i just noticed😮

• @mad face I actually looked it up. She just flips the screen after she’s done recording

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

• Like bro ..lmaoo

• this is only true for horizontal asymptotes right? these rules wouldn’t apply to the vertical asymptote at all

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• why lim exp negative X when x go to Xinfinity is +infinity

guys let us reach her 1M subscribers.

• U r too cute I sometimes get distracted by your smile :):);)

• OMG, she feels the mathematics!!!

• I am just getting lost in her eyes…

• Ok Nancy nice going great job and looking cool

• Nancy the Math Princess who rescued me from an F in calculus

• Same here.

• Shoutout to you❤✊🏾

• how does she write like that

• Finally found a video that relaxes you and gives you a headache at the same time 😀

• Thank you 💕

• I remember when she was barely starting out, because that was the first time i failed calc….

• 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!!!

• Youtube commentators have no limits!

• The best mathematician
God bless you

• Thank you, Nancy!

• Hey love from India

• Thank you ❤️

• the utter joy when i see nancy in my sleeep

• Who are the 72 people that disliked this at the time of writing this comment? I need to have a word with them… >: )

• The most beautiful MATH teacher I’ve ever seen.😍

• What could I say more than thanks? Thanks a lot

• you are very good dear you just make it simple for me.Thank YOU Nancy

• How can anyone concentrate?

• 0*infinity=¿

• SUMMARY: Anything over infinity = 0.
For those who have test tomorrow and don’t have 30 minutes. You dont need to know anything else.

• Nothing else? You can also get inf, neg inf, undefined/unbounded, etc. Or ANY ratio! You still have to apply logic, and if your professor wants a REAL i.e. mathematical proof, yer boned.

• If she takes classes, no guy is Missing a single lecture

• Wow! This is great! You are the best mathematics teacher ever

Lim X–>+inf x^2ln((x+1)/x)-x

• Thanks mam…love from india

• i am very grateful to you
thanq

• thank you very much

• Love u <3

• I feel very calm and relaxing watching her videos. PLUS, I might be even able to learn something.

• Does anyone know how to solve lim x→0 1/(2+3/X)?
And with the same equation as X approaches infinity

• I did it already but I will rewrite it

• lim as X approaches 0 (1/(2+(3/X))

• @Franklin Okeke the answer is 0

• Did you just substitute or use the transformation rule

• Because sometimes L’ Hospital or transformation rule can give you a much accurate answer as substitution.

• I wish I were in high school or college now.

• Dont worry NancyPi… No one is thinking dirty when you say HUGE, Large, or big….. buuuuuuuuuut the hand gestures on the other hand…. lol those probably more so…… it was rather quite cute watching you holding back your laugh. But thank you for the video it helped me find a answer to my problem…. along with some entertainment.

• Can’t Concentrate and Don’t know where to look at

• 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!

• Wait hold up for the algebraic way of solving limits at infinity, 0/2 is undefined right??

• That is equal to 0

• The limit of her beauty is does not exist.

• Lets hope I pass my Precal Honors Final tomorrow! Thanks for the help Nancy!

• The math Gods are good. They have given us this beautiful and highly intelligent professor. Thank you!

• You look alot like my crush Jasleen kaur gill

• in exponential portion i think u left the 2 number….where is 2

• I could cry. Thank you SO much!!!!!! #STEMINIST

• Such a math geek!!!

• I need this kind of teacher in math class .. 😂

• Great video, however I would have added a tidbit that these limits find horizontal asymptotes, with your 3rd type N>D finding an oblique through long or synthetic division.

• ur hot

• ⴰⵢⵓⵣ ⵏⵎ ⴰⵡⵍⵜⵎⴰ 👍🏻💗

• lim -x^2 with x->inf would´t be +inf, once -x- is +?

• u look hot

• U looking like Norha Fatah you are so beautiful…

• Excellent! I love your videos. Thanks for sharing with us your knowledge.

• you’ve saved me, ty

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

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

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

• 24:00
sin(x)/x = 1 just in case if the limit approches to 0 ???

• Yes

• it makes me very angry ! what does infinity really mean? very large or uncountable/unending ?

• It’s a symbol for something thats unending, but it’s not an actual number

• 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!!!!!

• how many of you just clicked the video because she is cute❤🤣🤣🤣!!!!!!

• Thank you <3

• Thanks, I love the video so much that I click the bell notification option

• Your lecture is highly commendable. Thank You!

• I think she explains better than the others. Thanks Pi

• 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! 😂

• Wish I saw your channel before having my Calc class 💀 I died like 7 times during this topic

• you explain this great your face is kinda distracting tho

• i really love the way NANCY explains

• tnk lot mam

• infinity*0=undefined…. why did you write 0 at 27:04 ??? I am confused

• Most beautiful educator in the world

• Nice

• 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 .

• you have boyfriend?

• 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

• Please do a video about the integration and differentiation of sin cos and tan pleaasseee, also wanna go out on a date ? Also maby some stuff about the chain rule

• Thanks mam

• Thankyou very much for your videos you’re a legend

• Beautiful, but you didn’t mentioned the type of the x is power to a value, like
lim (6^x-3^x)/(8^x+x^9)

• beautiful and intelligent.. WOW

• how am I supposed to learn when she looks like that

• you do a better job explaining this than my textbook thank you for your channel please keep making these videos

• At 5:45 to 6:15 we can explain this better

You explain Excellent… 👍🇺🇸 But you are confuse with the – & + signs, why they are here?
🇮🇳… we can make this more friendly to understanding

• Nancy, you are awesome!))

• I like the way you explain.. So relaxed.. Beautiful and talented teacher … Amazing way to teach.. Beautiful smile..

• whoever dislikes this is gay

• I wouldn’t recommend that you write equal to infinity for the first question or equal to negative infinity. It’s better you write an arrow to show that the limit approaches infinity or approaches negative infinity.

• I think she has to write backwards for us to be able to read it

• 27:49 My reaction when i realize i have calculus exam tommorow!

• I am feeling like crying!!! Why I find her today not before!!! OMG!!! Thank You Sooooo Much Dear<3

• I like this glassboard studio🤩

• There is a great pleasure in madness which only madmen know.

• Beautiful knowledge sharing. I always enjoy your explanatory steps. Big thank you

• Very nice explanation ma’am thank you so much 🙏🙏

• keep it up

• how can i focus on your lecture when all i see is your pretty face(“,)

• NancyPi, can you do a video on infinite series and convergence please?

• U r my personal teacher, u rock

• Helped so much, thanks!!!

• Hi ms or mrs Nancy.nothing to say but you top. really👍

• Hi, teacher

• You’re a life saver. Thank you so much for your content.

• she is fit ! awesome

• Very beautiful, fascinating and attracting lady.

• plz help me …
lim x—》3 (x²-9)/(x-3)

• Your videos are amazing. I am learning so much. Thank you so much for taking the time to explain math principles and laws with such care.

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