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.

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.

[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?"

[Ralvy]

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

[bonelesspizza]

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

[yeny]

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.

[ferreiraronald]

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.

[camd]

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!!!

[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!! :)

[chriswilliams]

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! :)

[hagaracherrat]

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.

[matthewanderson]

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.

[kellyjia]

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

[daviddial]

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

[bensmyth]

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

[nicholaswilliams]

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

[nadineamr]

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!!

[diana]

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! <3

[duncamacintire]

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

[joane]

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

[crazyxkid]

She explained how to find derivatives exactly as my Calculus teacher explained today





I wonder if he saw your video before class.

[cmejia]

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