Limits at infinity (KristaKingMath)

► My Limits & Continuity course: www.kristaking...
Learn how to find the limit of a function as x approaches infinity. In other words, if we take x to be infinitely large, what value does the function get close to? To figure this out, you'll need to use algebra to simplify the function, then use L'Hospital's rule several times until the function is simplified enough that you can evaluate it at infinity.
● ● ● GET EXTRA HELP ● ● ●
If you could use some extra help with your math class, then check out Krista’s website // www.kristakingm...
● ● ● CONNECT WITH KRISTA ● ● ●
Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student-from basic middle school classes to advanced college calculus-figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: www.kristakingm...
FACEBOOK // / kristakingmath
TWITTER // / kristakingmath
INSTAGRAM // / kristakingmath
PINTEREST // / kristakingmath
GOOGLE+ // plus.google.co...
QUORA // www.quora.com/...

[@isaaccraig3020]

This is a good way to get the concept but just for the point of making this easier you should explain that you only need to use the highest degrees and can remove everything else so in this example sqrt(9x^6/x^6)= sqrt(9)=3 provided that you make sure to expand or simplify correctly it works on any function if the highest degree is on the top but not the bottom the answer is infinity, if it is on the bottom but not the top the answer is zero, if the highest degree is on both the it is the ratio.

[@JustDave87]

There is a really cool trick to this just by looking at the powers in both the numerator and the denominator, which I bet you already know but I don’t want to spoil the surprise for anyone. Shortcuts mean nothing if you don’t know "why it works." Also, great job as always. I’m using your videos to keep my mind sharp for Calc II in the spring. Thank you so much for the explanations.

[@jsgames6843]

How'd you do in cal2

[@drylandthewanderer7026]

This one is easier if you divide top and bottom by x^3 rather than using L'Hopital. I was actually looking at this same exact problem this morning. Don't forget to change x^3 to x^6 when you factor it into the radical. It's just limit laws after that.

[@kristakingmath]

You could try to multiply both the top and bottom by the numerator, yes. I'm glad you like the videos! No, I didn't get a degree in math, I just tutored it and really like it! :)

[@bonganipeter9433]

You're really good at math. Doesn't matter whether you have a degree in maths or not. Thank you.

[@kristakingmath]

Dividing everything through by the largest degree term is the shortcut way of doing these problems, yes! :)

[@dylanwade2287]

Thank you so much!!! Working full time and taking summer classes is not pleasant! I didn't have a book during the first week of class, so this video has absolutely saved me.

[@kristakingmath]

You have a lot on your plate! I'm so glad this helped!! :D

[@norakau]

I really really benefited from your amazing lesson. I am a non English speaker and i had difficulty to understand from my prof but with your speed pace of teaching is very helpful. Thank you very much.

[@kristakingmath]

+Norah ALmasoud You're welcome, I'm so glad it helped!!

[@101dostoyevsky]

Thank you very much for the great work. I can honestly say you do a better job at explaining this stuff than my Calculus Professor.

[@kristakingmath]

Well thank you very much!

[@kristakingmath]

I'm so glad you like the videos! I'll continue to add more videos about applications of calculus as I can. Thanks, and good luck with your PhD!! :D

[@punkyhippo]

I used a different technique and solved this in many less steps and directly got 3 as my answer without any necessary cancellation...

[@akashk91]

Hi Krista. Your Math videos are phenomenal. Even more than these I appreciate the positive energy that you have around you. You're so lively and full of vigor that it inspires me. I am learning more than just math from you. Thank you ! :)

[@kristakingmath]

Awww thank you so much! I'm so glad you like the videos and that you're really enjoying them! :D

[@trendingbiz2654]

You just used the loooong and complicated solutions.

[@stephanieng9976]

wow that was extremely helpful. I learned more in 10 minutes of your video than 30 minutes of sitting in class.

[@kristakingmath]

+Stephanie Ng I'm so glad I could help!

[@3lsid]

Your're welcome! I thought I would help you, since I get so much help from these videos and besides, you deserves more exposure/views. Thank you.

[@QuickLieutenant]

so so sooooo very helpful. I learn more watching kind of videos than I do in my college class

[@kristakingmath]

i'm glad it's helping!! :D

[@kristakingmath]

I'm so glad I could help! :)

[@someone20ify]

Thank you. I like how you spend your time to answer all the comments!

[@gouravsharma4296]

Thankyou so much sister... You have won hearts of many students by your great work.. Am constantly watching your videos they are just wow... It's all your blessings that i am doing this much great in maths... Really thankful sister... Really mesmerized by your voice.. I know that this is an old video but still marking its presence.. You have changed my life.. From a very average to one of the finest.... Truly truly greatful..

[@kristakingmath]

I'm so glad that the videos have been helpful, Gourav! Thank you so much for letting me know!! :D

[@Ciukyexacta]

I agree, there was no need to make all the steps that were done in the video. it was enough to bring out the square root of 9 (3), and then prove that all what's left inside the root converges to 1, making the general limit equal 3.

[@XKlusively]

Wish you were my calculus professor . you explained this way better than my prof. Now at least I know what to do. Thank you much.

[@kristakingmath]

you're welcome, i'm so glad it helped!

[@kristakingmath]

Love you too! :) Glad you like the videos.

[@kristakingmath]

They're usually taught really close to each other, yes! :)

[@hajverycomputers7315]

I like your videos by the way.thanx a lot....its my first year at limit,continuity and calculus.And,m lovin' it..before classes,I watch ur videos..they really help.

[@ahcenedahmane5130]

i've found 3 directly without cancels ! you just do the fog(x) with f=sqroot of x and g=rest you get lim g is 9 and replace it with sqroot of 9 which is 3 ♥ Thank you ♥

[@tronghai55]

Krista in real case solving problems in complex maths if you can not understand the structural complex problem of an differential complex maths integral you cannot determine the differentials limits of your research in limits research fractions. Like I have said commun square functions or polynomials functions will lead us nowhere but you have to learn those basics maths in order to understand complex maths order. Only differentials polynomials complex maths can have an opening in the complex maths analog world. It's a whole new world out there. The limits had always its laws but how to structure it in a complex maths limits analog complex problem solving is another stance. That's why the applied maths will be the next generation.

[@ilaydagenc412]

Thank you so so so so sooo much madam! I think I will pass my math test tomorrow because of your great help! You explained it so clear. Thank you very much! Much love from The Netherlands! (sorry for my bad english)

[@kristakingmath]

You're very welcome, I'm so glad it helped! Best of luck on your test tomorrow, I hope you do great!

[@Slywolf1992]

Can't you just get to the step where you have a radical in both the numerator and denominator and when both of their leading coefficients are 6, then take sqrt(9) / sqrt(1) to get the answer of 3?

[@tronghai55]

The old school classics maths is for solving simple equations but on world problems of complex datas. Cause the datas to solve complex problems has to be applied advanced maths and not the Pascal reasoning. Advanced maths has nothing to do with pas Pascal reasoning solving. It's a another stance of complex problem solving approach. That's why your lecture is somewhat right but only in your head.

[@marietesdameg9259]

I know that the answer is 3 after I saw the function.. But I watched the whole video to see if I am correct. Very nice Solving. It helps me how to solve. :) -son of this account

[@dchurch2012]

Excellent video! I was looking for a video on L'Hopital's rule; Good example problem too.

[@kristakingmath]

Thank you very much!

[@Kalo953]

Hmm, isn't this a more complicated way for a solution. Isn't it easier to use L' hospital rule and then take the x^6 outside brackets in both nominator and denominator and then cross them out and simplify?

[@judahlancer]

thank you !!! you are the best on the net!

[@ThePinoyMamba]

wow. well done. I'm impressed... very clear... I did find it very helpful. thanks!!!

[@TheDiederikdehaan]

I think this case is also solveable without L'hopital's rule it's just more convenient to do so. Because in both cases you search for the polynomial of the highest degree (e.g. 2) and devide by the parameter in that polynomial (x^2 ), right?

[@crystal_snow875]

when this is the actual homework problem you needed help with.

[@crystal_snow875]

just kidding, im not allowed to use derivatives yet :(

[@iEmbraceU]

Crystal Bellus yea I am in the same boat lol. But it's like a preview of what's to come

[@gizmo_therapy]

very nice, thank you Professor King !

[@CarlBelly]

Thank you. Great video. I tried it and it worked. Although, I did not factor out a 6. I continued to take the derivative until I was able to cancel out an x^2. What are the different situations to use this rule? And what is the benefit to using this rule instead of taking the horizontal asymptote to find the limit as x approaches infinity?

[@kristakingmath]

i'm glad the videos are helpful to you. thanks for saying hi! :)

[@joaopedrofairbanks8888]

Thanks for the video it helped a lot, I guess it would be more simple if you have only divided the square root of 9x^6(= 3x^3) to x^3, though. Sorry if I mispelled or forgot to say something.

[@JossinJax]

Hi Krista. I appreciate the vids and hope you keep producing :). Quick question on this one: what was the function of putting the limit inside the square root? How did that have any effect? Because the end result was as if it was outside the entire time. Thanks.

[@kristakingmath]

Hey Jossy, As you can tell from the final answer, moving the limit around doesn't change the value of the expression at all. It just makes it a little easier to see what the math should be. Having the limit under the square root lets you take the limit of just the rational function, without having to think about or worry about the square root until later. But the answer still comes out the same in the end. You certainly don't have to do that step if you don't feel like it helps you solve the limit. :)

[@Matlockization]

Fascinating.

[@TSBooger]

man i love you for making this videos!

[@kristakingmath]

Awww! Love you too!! :D

[@Bistromathematic]

Thanks a bunch for making these videos!!!

[@kristakingmath]

I'm glad you like them! :)

[@kaustavphukan3951]

You could also have divided both numerator and denominator by x^3,which would have becum x^6 under the root,leaving u with lim of x tends to infinty [Sqrt(9-1/x^5)/[1+1/x^3] and applying lim would have given Sqrt(9/1)=3 straightaway..l'Hopital just makes it long.

[@tronghai55]

Krista like I said your functions to infinity is only a solving of commun standards of an equation of simple maths to somewhat infinity. But in real life advanced complex maths incase of special relativity or else you have to go to applied complex maths to deal with the integrals of complex polynomials maths shape shiftings to determine the complex of the conundrum of the century. Each complex maths solving is different of status to which you have to adapt the different reasoning of approach in order to structure the order inside the order to determine the complexity of the problem especially in the fermat conumdrum. And we both know that mathematicians want to put their name in the order to solve this conumdrum for money and fame by spying. Still your teachings is interesting to solve the logic of a commun ground so call logic. But in special relativity space approach this logic has to be deal in a another complex complex problem to which the resolving attitude takes time and effort. The infinity has its open unlimited limits but in real case problem if you can input the structurals logic of an évolutive definition of the conumdrum limits you can determine the pattern of complex logic to which you can work upon. And college maths is only collège maths cause it cannot solve the complexity of space and time relativity.

[@happybananainstitute]

Great job explaining!

[@kristakingmath]

Thanks!

[@TheSwaroop95]

Instead of putting a square root over the whole rational function can we just divide everything in the square root ( the numerator by x^6 and everything on the denominator by x^3..... I assume its the same thing right?

[@usama57926]

2:53 Taking square of denominator will make it positive even for *negative x values* . So it's changing the function www.desmos.com/calculator/dqrqwwpyl9

[@kristakingmath]

you're so sweet, thank you! :)

[@kristakingmath]

Yep! I just wanted to show people how to prove it with algebra. :)

[@kristakingmath]

you're welcome! i'm glad it helped. :)

[@yijiasun7495]

As one girl calculus teacher to anther, you should take off this video, it makes us un-creditable for math teaching. you should have got your answer already at your first 5 min.

[@mahamedhussein7216]

i agree.

[@MATHEUSALVESLIAO]

That works well, but can't you just multiply the function by x³/x³? Given that x³^2 equals x^6 things will just cancel themselvelves and the calculation will be done in one line. At least that's how I normally solve this kind of problems. But it's always interesting to find different ways to solve the same problem!

[@stepap7]

i think L'Hospital's rule is not applicable here cause you need something like ∞/∞ and not (∞ - ∞)/∞ which is undefined/∞ either infinity you use (+ or -)

[@MahmodoVX]

why u mixed the limits with the derivative? is it possible to answer that without the derivative?

[@aliasour30]

this is so complicated you don't have to do all this work. you simply need to look to the lead terms power in the numerator and the denominator to evaluate the solution. in this case they are equal, so you just need to divide 9/1 under the square root and the answer going to be 9 under the square root and this is equal 3

[@masons50]

Would the answer still be 3 if x were approaching negative infinity?

[@gaymo69]

what happens if you have a limit for a fraction like: [(1/2)^2 + (1/3)^2n] / [(1/4)^n +(1/sqrt(2))^n] Where the power is n

[@SonatinaClef]

Why didn't you just get for of the square root on the top by multiplying it by that? Am I wrong for thinking that's another method. While on khan I came across your videos and now I'm hooked. I know you said you studied psychology but did you get an AA in math also?

[@Sniperoidz]

Can't we just assume intuitively that, since the denominator and the numerator essentially have the same degree, the answer is simply the square root of nine (which gives your answer, three)?

[@jessmac1893]

This makes sense, but seems like a much longer way than i learned in the book. Instead I multiplied 1/x^3 to the numerator and denominator. The denominator results as 1 + 1/x^3. For the numerator I changed 1/x^3 to 1/sqrt(x^6) and then multiplied. ending up with a numerator of sqrt( 9x^6/x^6 - x/x^6). Solution is sqrt(9)/1 or 3. Did I do it wrong? Or is there a reason your method is preferred over the simpler one?

[@sail2byzantium]

Okay. You've lost me, starting at around 5:03. If I may air a minor frustration: I'm watching your videos in order as presented to me via KZbin, starting with the Foundations of Calculus (with which I was already set, but thought to review it anyway) and now, Limits and Continuity. I have been watching the latter in order as well. And none of these deal with derivatives, which I understand is another set of video--shouldn't something of that been discussed before this? How 9x^6 + x becomes 54x^5 + 1 or how x^2 + 2x^3 + 1 becomes 6x^5 + 6x^2 has left me baffled. It's not the difference quotient, that I can see. Should I have watched the set of derivative videos first before these on limits?

[@kristakingmath]

Sorry for the strange order. I usually try to keep the topics in a relatively logical order. This problem is applying L'Hospital's rule, which is a tricky subject because sometimes it's taught with limits, and sometimes it's taught with derivatives. I didn't want to leave it out of the limits playlist, in case people come here looking for it, but if you're watching the videos in order, it's going to trip you up. Sorry for the confusion... in general that shouldn't happen very often if you continue through the playlists. :)

[@TheSwaroop95]

do you learn this right after the intermediate value theorem?

[@daniloorbolato]

Hello Krista and thank you for your very helpful videos. One question. Could you use the conjugate "trick" and then start to take the derivative of it?

[@kristakingmath]

Yes, you could do that if you wanted to.

[@daniloorbolato]

Great. I found your way a little more "cleaner". Anyways its good to know. Thank you!

[@richardbisk8115]

What would the limit be if x approaches negative infinity?

[@user-gh4zs4sj9y]

Well thanks a lot for that But you could solve it easier Since the 3rd minute Thanks again for this

[@SonatinaClef]

Also as a math major I think I want to start making my own videos. Not these type yet because I'm only in Calc I right now lol

[@sophiedotdennison]

thank you !

[@hunterxhunter7090]

Thank you so much

[@kristakingmath]

Any time, Hunter! :)

[@rashadallahverdiyev9877]

by the way thank you for the videos, they help a lot !

[@zcruz123]

You made it more complicated then it needed to be...

[@kristakingmath]

hi new friend! :) I'm so glad I could help!

[@deephillips875]

I'm trying to write an essay on important current applications of infinity as it pertains to calculus. I can't find any information anywhere. Do you have any ideas or suggestions?

[@kristakingmath]

I don't know how physical or real-world your applications need to be, but in calculus you come across infinity when you're dealing with limits (precise definition, infinite limits, limits at infinity), with improper integrals, and with infinite series. :)

[@iml2492]

um isn't there an easier way to do this? my calc teacher showed us to just take the number with the highest power from the numerator and the number with the highest power from the denominator and just solve. so in this case it would be square root of 9x^6 over x^3 and that ends up simplifying to 3x^3 over x^3 and that simplifies to 3.

[@dywing]

haha ur my new best friend!!! u help so much!!!

[@kristakingmath]

You're very welcome! :)

[@lalengbarce1338]

great explanation👍👍👍

[@kristakingmath]

Thanks! I'm glad you liked it.

[@JosephHoward]

Well explained! :)

[@kristakingmath]

thank you!

[@4622201]

It 's always nice to see a girl (also nice), expert in calculus!

[@baghdik18]

do u know how to do lim sqr of x+5 - 3 / x-4 x--4 the answer is 1/6 but in don't know how to get the answer thank you

[@kristakingmath]

You're welcome! :)

[@3lsid]

There is a youtube Nextup competition for youtube educators. I thought you might be interested. I saw khanacademy retweeting the link to it.

[@kristakingmath]

Thanks! :)

[@tronghai55]

Krista the lessons of limits is only an extension of classic mathematics but in real life problem engeneering solving you cannot applied those old methods to comply with the high standards of applied complex mathematics. Still your lessons are interesting for basics maths only not in the new science complex engeneering in artificial intelligence. Of course as a student you have to go through those basics cause the basics is an opening to more complex solving. I am still not able to do what you do because of lack of functions memories. But later on you will see in the future complex applied maths will be the new age in artificial intelligence. Cause the classics functions can never solve the complex engeneering maths in computing sciences.

[@grahambest5712]

you lost a -1 between 4 and 5? why did that happen?

[@kristakingmath]

thanks!! :D

[@grahambest5712]

like step 4 and 5? I'm not working on calculus in school so it might be right, but, you lost me their. even the word darivative? what is the darivative of something?

[@indian_scouser_ynwa]

nice video once again.hats off to u mam.where can i find your e-book mam?.eagerly waiting to see and solve.thank u

[@kristakingmath]

Send me a message here and I'll send the ebook right over! www.kristakingmath.com/contact-me

[@asus9596]

agree

[@codosacho5924]

I have a question how can i quickly figure out the derivative of function?

[@kristakingmath]

+Codo Sacho Learn your derivative rules, like power rule, product rule, quotient rule, and chain rule.

[@logosfabula]

Hi Krista, in which video did you introduce derivatives? I must've skipped it.

[@kristakingmath]

logosfabula Right here: kzbin.info/www/bejne/eoO7fJqveNCKg6s

[@logosfabula]

integralCALC Thank you! I was following a different playlist.

[@maryelizabethgore9913]

What program are you using to write on? I like the look of it.

[@kristakingmath]

+Mary Elizabeth Gore It's called Sketchbook.

[@rinocastaldo2329]

non c'è bisogno di disturbare il marchese De l'Hopital; con x che tende all'infinito e due polinomi di ugual grado il limite è automaticamente uguale al rapporto dei coefficienti delle potenze di grado massimo : 9/1=9

[@onlyCreativity]

Wow, that was complicated. Why can't you just say lim (Sqrt(9x^6-x)/(x^3+1))= Sqrt(lim ((9x^6-x)/(x^3+1)^2)) That way you can ignore the outer square root and calculate the lim of (9x^6-x)/(x^6+2x^3+1) wich is 9. Now take the root of that and the result is Sqrt(9)=3. I think that works with any root.

[@jessietorres1387]

Very helpful!

[@kristakingmath]

i'm so glad it helped!

[@jessietorres1387]

Do you have anything in proving a function is continuous?

[@kristakingmath]

Some fall through the cracks, but I do my best. :)

[@wanreaper]

Thank you ! :)

[@kristakingmath]

you're welcome!

[@mahmoudsamyessawy]

thank yoooooooou , it is helpful

[@tronghai55]

Krista the future will be complex maths analytics structural analog design. It's will be analog computing design or else. And the old school of maths cannot survive in the computing analog datas systems for the simple reason analog has to have a dynamic analog base system for computing science. And solving complex system you have to understand what is standing behind the reasoning of complex datas before you can go to an computing analog system for fast datas. Life is not fair but how life is.