# How to Find Any Limit: Part 2 (NancyPi)

197,652
Published on May 18, 2018 by

MIT grad shows how to find the limit at a finite value with a square root, (sin x)/x, or absolute value. To skip ahead: 5) for a SQUARE ROOT in the numerator or denominator (to RATIONALIZE by multiplying by the "CONJUGATE"), skip to time 1:14. 6) for a limit with something of the form (SIN X)/X, skip to time 5:38. 7) for an ABSOLUTE VALUE in the limit expression, skip to time 14:45. Nancy formerly of MathBFF explains the steps.

5) For a SQUARE ROOT in the numerator or denominator: If you try plugging in the value for x and get a 0 in the denominator, and you cannot factor, get a common denominator, or expand to simplify the expression, then if there's a square root in the numerator or denominator, you can try MULTIPLYING by the CONJUGATE. For instance, if you have sqrt(x+1) - 3 in the numerator, you would multiply both the numerator and denominator by sqrt(x+1) + 3 because the "conjugate" just means a two-term expression with the sign flipped in front of the second term. This is a trick or technique that helps simplify because when you multiply out, or FOIL, the numerator you will get terms that cancel. It is best to leave the denominator factored, rather than multiplying out the terms since a factor is likely to cancel. Once you simplify by multiplying on top, combining like terms, and canceling any factors from the top and bottom, try plugging in the value again for x to get an actual limit value.

6) For the form (SIN X)/X in a limit expression: If you try plugging in the value that x is approaches, and you get 0 in the denominator, if your limit expression is something of the form (sin x) over x, there is a trig property that you can use to simplify. The property is that the limit of (sin x)/x, as x approaches 0, is equal to 1. If your expression isn't exactly (sin x)/x but instead has something like 2x or 3x inside the sin function, like sin(2x) over (4x), you can use the same property but first have to rearrange the expression in a way that matches what you need, as shown in the video. NOTE: Be careful not to confuse this trig property with another, very similar, (sin x)/x expression for when x is approaching infinity. That property states that the limit of (sin x)/x, as x approaches infinity, is equal to 0. Check out the video on limits at infinity for an explanation of how to use that expression.

7) For an ABSOLUTE VALUE in your limit expression: If you try plugging in the value for x and get 0 in the denominator, and you have an absolute value in your limit expression, you will probably need to re-write the limit expression using the piecewise definition of the absolute value function. You will then have an expression for the left-side limit and one for the right-side limit. If you evaluate the left side and right side, and the numbers agree, then that is your limit value. If the two sides do not have limit values that agree, then the limit does not exist.

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

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• Finally

• Just in time to prep for my Calc final! Thanks for the educational vids 🙂

• NAAANCYYYY HIIII

• wooohooo! Yayayayayay! Thank you!!

• Nancy’s on fire 👍 the best ever

• GooD work.

• Thanks again Nancy, I’m a limit ninja now due to your help! You’re the best!

• Does anyone else watch these because of her amazing voice? She’s like the Bob Ross of math

• I watch her for scientific purposes 😉

• Just kidding thats gross and not cool

• morgan freeman

• I don’t take any math classes but I just watch her videos because she’s cute 😂😂

• … and her smile

• Yayyy!!!!! I’m so glad you’re here!!!!

• Why am i paying international tuition when this is free but both of the contents are the same?!

• uzair mirza revolution is coming

• I am paying international tution too !!! 😡

• Same reason why anybody pays for any kind of education so you get that certified price of paper for virtue signaling on job applications otherwise they don’t care if you were self-taught through other means unless you somehow obtain work experience from being an entrepreneur at which point, why work for anybody else when you can be your own boss. Harvard is now offering free online classes too but you have to pay to get the “credits” for the class.

• Could you do some on where all these formula is used and how its applied in life. NZ.

• Can you go over some advanced topics? Like some hardcore proofs involving groups and symmetries or vector calculus?

• Carl Johnson yesss or how to graph

• *Love the videos! DON’T GIVE UP NANCY*

• Hi Nancy u are beautiful and thanks for calculus and plz cover all topics of limit step by step…

• Glad to see u back

• Good work is always appreciated

• And it should always be promoted

• thank u♡

• graphd are much needed (like if u agree) plz do a video about them u know those graphs that u use limits to find where they are that looks like snakes 📈📉i still have trouble with them..♡♡♡♡

• Please cover lim at infinity & lim at infinity for integration.

• These videos are just so relaxing. I watch even when I know the content 🙂

• I know what is so relaxing 🙂

• Hi Nancy, great videos! For the absolute value example, why do you show that x =2? Since you can never divide by zero, the domain of this rational function should “exclude” x=2, correct? However, in your example, the piecewise graph shows a “solid dot” at the point x=2. I think it should be an “open circle” from both the left and right limits. Likewise, the piecewise function should not contain x=2, agree?

• Quick suggestion, it’s great to see your face, however at times it can be hard to read the writing due to low contrast between the background and the text color.

• Nancy, I am so thrilled you are back. You were missed immensely.

• Nancy, Do you have a video where you find the area under the graph along with slopes – I guess I am asking for calculus examples?

• Instead of multiplying by the conjugate, why didn’t you just do L’hopitals rule?

• I would hav asked the same question if she used l’hopital’s rule 😁

• hi. can you make a video on laplace transformation

• It was awesome tnx…Nancy

• What about RELATED TIME RATES FOR DERVIATES AND INTEGRATION CAN YOU HELP WITH THAT PLEASE !!

• What about bounded integrations !

• Area and volume of bounded integrations

in engineering field where we really apply math concepts ?

• Raghupathi K 😂😂😂😂

• this was really helpful thanks sm! Can you do a video on integral applications? Like finding volume by rotating around an axis, area between curves, and using cross sections? Thanks so much for your help

• lmao she has a weird knack of uploading these videos at the most convenient times

• Why is lim x–> 0 sin 2x/2x = 1 ? It has to be lim 2x-> 0 sin 2x/2x = 1

• Because on X–>0 ,x can be any parameter like (2x) or (t) or (3t)

• love your video ,well done ,

• Hey nancy please teach us some applied mathematics for physics. BTW nice work👍👍

• Hey Nancy, can you post your videos of logarithms? They were amazing and I really need them for my maths class. Thank you for all the videos, they have been so helpful.

• Hi Nancy i just found your channel and im loving how well you explain things. Please dont stop uploading. Especially calculus videos. You make math easy to understand! I start calculus for the first time in the fall and will study it till then 🙂

• Nancy, you are my crush!

• Imagine a NancyPi ASMR version…

• she looks like someone that would be in a shampoo commercial.. XD

• I just need to find 4 or 5 women like this, smart and hot!

• True that

• Why not just invoke L’Hôpital’s Rule and skip all the algebra?

• Tallis Dawson i was taught derivatives before limits specifically so i could use L’H instead of all the pointless algebra

• @Paul How are you taught derivatives before limits when the derivative IS a limit? That makes no sense at all, you literally just memorize a bunch of formula and rules of derivation then, and don’t really understand a thing about it.

• InStinct because i went back after limits and covered the formal definition of a derivative. i have a masters in applied calculus so i don’t think i just memorise formulas…

• @Paul Well yes, you went back, but I think for most people it would get damn confusing. At first you definitely just memorized formulas, because without being able to calculate a limit you can’t calculate a derivative. It just makes a lot more sense this way, and there is a reason why this is the way it’s being taught at almost every school and university that I know of.

• InStinct you’re saying i got taught that way, i didn’t. it’s simply just easier to do limits second

• So good videos that there are no dislikes even after 2.1k views 😯😯😯

• What equipment do you use?

• Nancy thankyouuuu very much for all the videos

• Want some inverse trigonometry videos

• Your videos are so awesome and helpful! I would love if you could cover how to do approximate integration and the area of a surface of revolution since I am struggling with those concepts currently.

• Math Megan Fox!

• Like a… ninja… 🙂

• She is left handed ?

• Just in time for my Calculus class and test on limits. Thanks so much Nancy!

• You know, if you joined steemit.com and our STEM community there, you could make quite a bit of money from your videos – maybe itll inspire you to start it up again? A shame to see it stop!

Feel free to msg me for more info at hostmorker@hotmail.co.uk – i can help set you up and pretty much guarantee income and a great community of like-minds!

• I love how you organize everything into blocks like this. It can be overwhelming having the different techniques individually floating around my mind and you make it so much easier :’)

• Hello I was wondering if you would be working with differential equations I will be starting my class next month and learn more from youtube videos than class. Thank you!

• Hey! Fun to find a fellow MIT grad here. Just started my channel. Are you still making math videos? This whiteboard and fast forward effects are cool. Keep up the good work =)

• Nancy is so hott.

• u should do asmr videos

• Can you also discuss double angle, half angle, triple angle, product to sum and sum to product formulas? Those are the formulas I have trouble memorizing myself.

• NANCY’S ON FIRE, YOUR EXAM IS TERRIFIED, MATH IS COMING HOME

• why can you simplify (x-8)/(x-8) in the limit if x goes to 8? that’s like simplifying a « 0/0 » situation

• I love you so much………..I am one of the millions who love you so much……….thanks a lot for your great videos…..God bless you forever.

• nice work,Miss nancy plz make a video lesson on probability ,factorial and statistics….

• I figured out how you write like that. You are 100% dyslexic, right?

• Thanks a lot NancyPi <3

• good

• What if we get infinty on both left and right limits ?

• I drank every time you said “OK”. Yipes!

• you are a godsend

• ?

• Big love Here from Egypt for your ( EFFORTS ) I really respect you Nancy :))

• This helped a bunch! Thank you!

• Thank you Nancy, the best!!

• I owe you big time! I’ve gone back to school to finish my math major and go to dental school and I’m having to relearn all of my calculus (its been 8 years since I first took Calculus 1!). You are helping me so much! I work two jobs full time around school and never get a chance to go see my professors during office hours, but thankfully youtube is never closed! Thank you for your awesome contribution to the learning community =)

• love you <3

• Please do lectures on probability ans statistics

• Here to become a limit ninja!! Thanks yet again, Nancy!

• You’re a better teacher compared to my college professor 😭😍

• Third world schools unfortunately are not as good as American schools. No matter what anyone says. Wonder why some people think they’re so easy even though the US always has good tech that works. Someone is lying lol.

• College professors are souls from hell

• Hi Nancy … At 21:33 you have a little mistake since the function is not defined at x=2 so the point (2,1) must be white too just like (2,-1) does not belong to this graph

• yes at 2 a vertical asymptote is reached

• She’s absolutely beautiful

• “You’re going to be like a ninja….. but for finding limits at a finite value…”

• LOLLLL I was gonna post this exact same thing and I can’t believe it took so long til 2 days ago for someone to say it

• And she was like stifling her laugh

• Nancy can you please explain how did you got 3(square root x+1+3) twice?

• How did you came by 1/6

• I’m amazed by her ability to write backwards

• Really a big help 😊

• you’re awesome !!

• Wow I love your videos I wish you the best!! Your videos are so understandable and helpful!!!

• what about u-substitution or the long division factoring?

• Some log/exp limits examples would be great, we’re having issues at calculus with those ones.

• ARE YOU WRITING ON THE CAMERA ??? xD

• jajaja😂😂

• Super…

• Do you never use L’Hopitals rule? Is it better not to use this?

• I LOVE YOU

• can you make for me (x-8) to be the conjugate cause I did it and I found 3/16 is it answer

• Helal abla, thank you very much!

• you are blessed…keep up the good job

• hello mam do u hav any videos over fourier transform

• Finding limits and writing beautiful, readable text backwards on a transparent board have been on my skills-that-are-impossible-to-master-list for a long time. Now, of the two, only the writing-backwards-part remains on that list (together with writing readable handwriting in general). Thank you for an excellent explanation and presentation of the subject!

• She doesn’t write backward, the video is mirrored in editing. Her explanations truly are amazing, though!

• @James Allbee Aha, that makes sense, silly me :-). Thanks for clearing that out. It’s still a very interesting and entertaining way of presenting her explanations, like you say. I never believed that David Copperfield could really fly, though. Promise 🙂

• How is lim4xsin5x/sin4x * 5x/x = 5
Lim 4xsim5x/sin4x is 1 i get that but 5x/x isn’t it 0 ?

• the X’s cancel out

• Thank you mis Nanci

• I wonder how you write that way

• Nice video

• At 3:56 Well you could have made the example shorter by just using a2 – b2 formula! 😀 btw you are teaching amazingly n tbh u r the best! 😊

• god bless you…

• Gud

• this is helping SO MUCH!! Ap calc test tomorrow

• Hot

• Does anyone realised.. That.. She.. Wrote.. In.. The.. BACKWARD DIRECTION?? Or its just my eyes being wrong

• You should be showered with gifts, and men and women to give you pleasure!

• Great work Nancy!. I am trying to get a simplest way to express a calculated limit. Would I be correct to say that the limit (as calculated) is the value at which the X value cannot be? In other words, if the X reaches that value then the two points on the secant would meet at the same point (tangent) and therefore there would be a zero in the denominator, rendering in indeterminate or undefined?

• I’m Turkısh but I wanna prefer listen and learn maths in English. Especially this channel also her voice perfect

• Math question for you guys: does anybody know the limit to Nancy’s beauty?

• Ur so cute when u catch ur breath😂😅

• Thanks Nancy

• You are saving my life in calculus right now!

• At 13:50 confused me a bit but I think you only made a written mistake where you put 5x in the bottom(making it be 5x^2), being only 5. Am I right? (When u multiplied top/bottom by 5x)

• I understand…WTF?

• youre the only channel i come to for help with math. you have all the material covered, and explain it so thoroughly. thank you!!!

• Ninja analogy was horrible, stick to math lol.

• doyou have tangent?

• Why didn’t she tell the L’Hospitals method?

• Ok so I think I got it, she is writing on a mirror. Damn that took a lot of thinking, now back to math.

• For 11:20 can’t you just multiply the top by 2 to get 4x and then multiply both by 1/2. I did and I got the same answer just like the previous question

• Please do a video on epsilon-delta proofs.

• Best of the best

• You are awesome!!!!!! I learned more with you than my professor about limits. Good Job Nancy

• Thanks

• BTW, Nancy please keep using algebraic methods because the L’hospital rule or whatever is not accepted by ever professor, at least at the start of the semester. Technique matters.

• hey lefty let me remind you im  also lefty 🙂

• l really like your vedio..that can help me much in calculas

• Sooo… Practically, in absolute values, usually limits do not exist??? Not all the time but usually

• 13:38 is the worst explanation ever

• Pretty and beautiful almost as much as the number of digits of Pi. I’ve fallen in love with you as fast as Light loves to run through the Universe.

• NANCY PLEASE DO VIDEOS ON CONTINUITY I REALLY NEED HELP

• make video on exponential and logarithmic function

• making my life of maths simple

• can you help me find the limit as x approaches zero for sin 2x over 5x. I tried the rule that you are using in this video, however, the workbook that I am using has answer key and it tries to explain that i should take the derivatives of both numerator and denominator then substitute x=0. Any help would be greatly appreciated.

• I want to say thank you for these videos on limits. It greatly helped me understand how they work and made the final exam an absolute breeze! Your videos are very understandable.

• By the power vested in me, I invoke L’Hôpital’s Rule to expedite the problem!

• 22:16 LOL Yeah right! Limits are my favorite. I like them almost as much as going to the all night dentist. They’re especially fun when absolute values are involved. I look forward to them almost as much as a trip to the laundromat. Thanks so much for going over them step by step so they made more sense. A few more examples and I’ll be able to teach this class.

• How to find you in my life forever Nancy, what can I do, answer please

• Yeah, she’s definitely going to answer this message. It’s not creepy in the slightest. lol.

• Very nice. Thank you!

• Actually use HOP is very more useful…
And
Sin2x=2sinx.cosx
U can use it also

• It might be more useful but more complicated.

• She has Gray Eyes <3

• Wow, Nancy, your smoking today, I must say. Whatever you’ve done, keep doing it. 😉

• who else thinks she cute and smart perfect combo!!!!!!!!

• I just melt when she explains something i dont understand.

• Thanks for the help on that! I’m in Honors Pre-Calc and was finding these complex forms of limits somewhat confusing.

• For absolute value problems, what would happen if absolute was in the denominator?

• What happens if the Numerator and the denominator both have absolute value?

• So smart and so beautiful!

• I genuinely came here to learn limits,
and now I feel like a ninja !! 😋

• Lim sinkx/x=k
x-0
E.G sin5x/x=5

• So when finding the limit using the absolute value, you got -(x-2)/-(x-2) = -1

Shouldn’t that answer be equal to positive one!

After all, it’s a -1/-1 ; A negative being divided by another negative….

Thanks Ms. Pi,
You’re beautiful!

• Nancy has a talent of making things simple to understand and remember! thank you!

• Beauty and Brains!!! She is not only helping me with math but i think I’m in love!! Lol!

• This channel deserves way more subscribers.

• Yo how the heck do you write so fast?

• at 4:00 you FIOLed instead of FOILed lol

• I correctly completed my first limit. Thank you! You’re a blessing

• She is prettiest than Megan Fox and has amazing math skills.

• Great work nice lecture

• doesn’t l hospital rule comes along with the topic of limits

• Thanks for the great video. This really helped me. 🙂

• Awesome explanation!

• Thank you , really you make this object very easy

• u cleared all my doubts.thanks. are u a professor?

• What if the absolute value is in the denominator?

• I used to listen to lofi while studying. Now I listen to Nancy.

• I really love your videos nancy it really help me a lot on my calculus subject.

• I need finding limits including sgn function and Greatest integer function, Indetermine forms for my JEE Exam… The ones you discussed are basics….
But thanks 🙏

• 😭 Thank u!

• I came at the video bcoz of your beauty and i watched it bcoz of your genuinity and credibililty

• How to write on that wall ….?????

• Your videos are very satisfying 😁 thank you

okay its visible but i mean dark marker would be perfect.😉

• hey its Satyajit from IIT , India …..would you like to make a video on limit for multi variable function?

• hi, I was abit confused about the first equation, your numerator is (x-1)-9, isn’t that -9x+1? but why did it became x-8?

• That tripped me up as well but it’s all about the parenthesis. (x+1)-9 = x-8. However, (x+1)(-9) = -9x-9.

• I meant -9x-9😅😅

• what happpens to the 9 in -9x-8 for 5:00

• I have some troubles with absolute type of limits😩😩

• Currently using these as a refresher for my Cal A midterm. Thank you a lot!

• you are beyond my teacher even.i understand more when its you teaching.

• Thanks! I’m gonna need this for business.

• Adjust playback speed to x1.25 and thanks me later

• Learning ang hardening at the same time!!!😉

• Oohh Nancy you can turn me on!

• I am missing the limits on which you have to use Taylor expansion :/

• loved it like every single one other…

• God she is something else I often forget I’m watching a math video

• How do you fast forward

• wow this is amazing, my lecturer only covered 3 topics out of 7 topics in calculus. so these videos really make things so simple. i rate them 5 stars (*****) thank you so much

• omg nancy your jokes are mainly what keeps me going lol

• A beautiful lady…teaching a beautiful subject. Just one query…at 14.06 shouldn’t the limit be indeterminate since lim as x approaches 0 of 5x/x = 0/0? Were you just trying to catch us out…or am I incorrect?

• Beautiful

• I fall asleep with her videos

• If all the teachers in school are like u students never bunk the class

• Thank you so much helped alot✨

• Please please please… stop saying “cancel out”. A puppy dies every time you do so. 🙁

Hyperbole,? Yes. But an important distinction in Maths; things subtract or divide out. They DO NOT ever cancel out.

• Please solve some more examples

• Can I just thank you for taking the time to edit in the fast-forwards? I’m learning it either way, but it makes it even more classy. Stay classy, Nancy Pi.

• i am sorry i watch these vids for youre beauty so hot

• My limit is 8 pints.

• how to find two limits in one function!

• At 0/0, l’hopital rule is easier!

• Nancy we need laplace transformation

• Hi

• Thank you

• oh wait, she’s a lefty

• This is the same video as part 1

• Thank you so much for your videos! Really great help for me in surviving Calculus 🙂

• You are a Bob Ross of math

• Where did -9 go after rushilizing

• I think there’s a method that’s still missing.
Fractions of two high grade polinomials
I mean:
lim x→-1 : (3x^3+8x^2+7x+2)/(2x^3+7x^2+8x+3)

• Hey gays😂😘 schön

• Saved again by Nancy talking about math with ACTUAL WORDS!!!!!

• You’re smart and beautiful.
Loved the video, thanks

• is the most confusing part for me.

• learned more watching this video than a semester and a half of university THAT I AM PAYING FOR! thanks so much

• marry me pls

• Such a clean explantation, am your fan

• Had anyone noticed that when Nancy was writing very fast, a slow sound was produced??

• I don’t agree with you about absolute limit
Bcoz if x is left approach to 2 but not equal 2 so ,x-2 =-ve,so the numerator and denominator -ve and the results is +1 also in second case please if not I need explanation from you

• Good Teacher ❤

• How did you get the right limit equals to 1 in the last problem¿ shouldn’t it be 0/0 as x approach at x-2/x-2¿¿¿

• Just wanted to say thank you thank you thank you

• Little dry jokes.

• 09:20
but the Θ = 2x
and bellow lim we have Θ = x -> 0
x = 2x ??

• Miss. Megan Fox of maths.

• There’s no limit. I want to find you and marry you. XD

• Why has nobody mentioned the fact that she’s been writing backwards? (That’s crazy)

• Thank you maam.. Love and respect from India.

• wish u wer my tr in my school days.

• Nancy I love you. I’d just like to tell you what I feel in front of those witnesses that wouldn’t expect to hear what I mean.

• Love from India

• 02:12 Trick :
Multiplying with the conjugate…
2-term x by the same, only the sign in the middle changes

• Thank you so very much for all your help!

• U ARE GREATER THAN OUR PROFESSORS ….

• Nancy, how are you memorizing the trigonometric rules + algebra rules…this is calculus 😭

• She explains very well, but for some reason I can’t concentrate

• Wait, is she writing everything backwards?