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 https://youtu.be/nJZm-zp639s

For HOW TO FIND THE LIMIT AT INFINITY, jump to https://youtu.be/nViVR1rImUE

Follow Nancy on Instagram: https://instagram.com/nancypi

Twitter: https://twitter.com/nancypi

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: http://nancypi.com

كيف تكتبين وما يطلع بالعكس

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

Sexy!

im not here to learn 🙂

Can you be my PRIVATE tutor?

Is she writing backwards!? Lol wtf

Wow, a beautiful mathematician!

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

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

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.

Damn! I am just focusing on her.

Thank you a lot!

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!

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

hi dear i have a confusion

thank you so much this was so much helpful

im sorry teacher i just can’t focus

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!!!!

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

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

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.

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

Thank you Nancy. You got me through Calc 1

Bend them. And move across.

I love you

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

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

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

شكلج مال نوم وزواج حرام تخربي بالرياضيات

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!

the limit is blowing my head with a gun

You are literally a GOD SEND TY <3

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

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

Thank you Nancy. These videos are incredibly helpful.

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☺👋

Very beautiful

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!

🧱

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

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!!

SHE SAVED MY ASS BRO SHE RLY DID THANK U SM ASJVBJSI FINALLY GET IT ASJBCSKZ

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

from which country are you bro .

@Aditya Ajay India

I got a feel with your language , same here.

mummy boli, “ye kya hai , maine kaha calculus ,

mom had some doubts “.

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

@Aditya Ajay nah im in 10 currently preparing for ioi

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

this content is awesome!!! very helpful

So sexy my dear Nancy ma’m!

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

Good

🥰🥰🥰

WHAATTTT?? Still searching youtube for math help?

Are the videos from 2006 realllly helping?

she is explaining limit, without defining an arbitrary function.

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

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

Excellent explanations, congratulations

Great man g

1:27

Good one

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

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

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?

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

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

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!!

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.

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…

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

Good❣❣

Nice …parabolas!

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

Thanks from Argentina in the very south of the world…

i wonder how i can use this in my computer programming

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

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

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

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.

nacy chan saiko

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? 🙂

oh i love math !!!

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

I salute you my dear sister

You are looking nice

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

mam please explain the definition of limit

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

YOU SHOULD HAVE WAAAAAAAAAAAAAAAAAAAY MORE VIEWS thanks alot

Best video on YouTube

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

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

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.🤨

CRAZY!!!😖

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 😚

So Smart and Cute Teacher.

How you write? I need like program

The only matheatics videos from which i dont get bored!

how do you write like that?

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.

You are very beautiful

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 !

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

Mam u r very cute…….

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.

Wooww! So pretty! 💕💕💕💕