What is L'Hospital's rule?

► My Applications of Derivatives course: www.kristakingmath.com/applic...
0:45 // What does L'Hospital's rule do?
2:36 // How many times can you apply L'Hospital's rule?
3:26 // An example using L'Hospital's rule
5:13 // When you can and can't use L'Hospital's rule
7:35 // What to do when L'Hospital's rule fails
7:51 // Two examples that avoid L'Hospital's rule
10:11 // Another example using L'Hospital's rule
L'Hospital's rule is so named because i was discovered (maybe) by the mathematician L'Hospital in the 1600's. While L'Hospital is credited with the rule, he worked closely with Johann Bernoulli, who claimed after L'Hospital's death that L'Hospital had paid him off in order to take credit for the rule.
But no matter who discovered it, L'Hospital's rule is a really helpful tool that you can use to solve limit problems. You want to use L'Hospital's rule when you evaluate a limit and the result is an indeterminate form, like 0/0 or infinity/infinity. For a limit that gives an indeterminate form, L'Hospital's rule says that you can replace the numerator and denominator of the original function with their respective derivatives. In other words, replace the numerator with its derivative, and replace the denominator with its derivative. Then use substitution to try to evaluate the limit again. Oftentimes, L'Hospital's rule will have simplified the function to the point where substitution no longer gives an indeterminate form, but instead a real-number answer.
And L'Hospital's rule can actually be applied multiple times. So even if you use it once, and then try substitution and you still get an indeterminate form, you can just try applying L'Hospital's rule again and again until you eventually do get a real-number answer. The only drawback is that there are a few instances in which L'Hospital's rule can't be used.
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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
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[@justinward3679]

The hardest thing about L'hospital's rule is pronouncing it.

[@harshvardhanbauddh2877]

Actually i got it now

[@CaptainSpock1701]

The hardest thing is that font!

[@usmanalam23]

I was told by my lecturer it is pronounced "Lopital's Rule" with a slight accent on the "o". Also it is spelled "L'Hôpital's" [French: lopital] but it would seem a lot of American classrooms have settled for L'Hospital. Poor man

[@stumbling]

Guide to French Pronunciation: Select 40% of the letters in a word at random and only pronounce those.

[@steveclarke1857]

*L'Hôpital No S.

[@Luhgooey]

Thank you so much! You explained it much better in 13 min than my professor did in an entire lecture.

[@ddahstan6876]

Not only the video is clear to grasp but I'm intrigued by a little brush of history, too! Love your passion and thumb nails! :-)).

[@josem.barquero604]

Thanks.....even though I am not taking my calc course anymore, I still love watching your videos. So easy to understand.

[@kristakingmath]

Love that! Gotta keep the skills sharp. :D

[@dannyshaw4057]

Really well explained. In fact this is a channel full of great knowledge and explanations!

[@osagiee.guobadia-secondytc4624]

Good information about the L'Hospital Rule while I'm attending Calculus in my 2020 Fall semester, Miss Krista King! : )

[@kristakingmath]

Glad it was helpful!

[@praptisalaskar6281]

Wow, you're amazing, ma'am. Thank you so much, keeping posting more! :)

[@kristakingmath]

Thanks, Prapti! I'm hoping to get back to posting soon! :D

[@RafaelRabinovich]

Thanks for the practical lesson.

[@stumbling]

"Lowsbietowel" "Lowsbie... Towel" "Lowsbietowel"

[@bhagatcharan1127]

Great explaination sister

[@rajendramisir3530]

Thanks for sharing three conditions where L'hopital's Rule will not work. Johann Bernoulli was a clever Mathematician.

[@mathematicaATD]

Nice explanation

[@Webusage1988]

Such a cool rule. Thanks for teaching it clearly.

[@kristakingmath]

:D

[@mohamedsaeed8911]

Thank u .. your videos are so helpful 👍👍

[@kristakingmath]

Thanks! 😊

[@khalilhodroj8936]

thank you

[@gooball2005]

The last example can also be solved by factoring the denominator twice.

[@fetauAdu]

No need for L'Hospital rule for it was factorable both the numerator and denominator using the difference of squares.

[@MMA_CAT]

you rock my world ( MJ's voice)

[@jayprakashsingh2034]

Thanks it helped a lot.

[@kristakingmath]

You're welcome, I'm so glad it helped! :D

[@lanvu9323]

thanks!

[@WagesOfDestruction]

At 8:31 you can only divide by (x-4) if (x-4) does not equal zero.

[@paulway7745]

Beautifully explained. Can you tell me what program you are using for writing? Thank you!

[@kristakingmath]

Thanks, Paul! I use Sketchbook Pro as the blackboard program. I really like it and would definitely recommend it. :)

[@paulway7745]

Thanks Krista! I will definitely be looking into it.

[@MackMillicco]

So... you got any experience in vector spaces properties? I've seen that your videos are mostly Calculus (which I appreciated during those classes) and I was wondering if you would ever think of branching out to, maybe, linear algebra. I'm looking at these matrix properties and I feel like I walked into an upper foreign language class.

[@kristakingmath]

LOL, hang in there! :) It's definitely something that I hope to do at some point, but I know you need the help right now.

[@sircourage]

You are a well organized teacher

[@kristakingmath]

Thanks, Manuel! :)

[@mansarawat2450]

Thank you

[@JoeyBartlett]

It was so useful that I thought there was something wrong..

[@manuelaidos]

I love this rule mostly when i need to calculate indeterminate limits

[@StarLord1994]

When else would you love it? Lol

[@Nursin-rg1ey]

Amazing 🤩❤️

[@kristakingmath]

Thanks, Nursin! :D

[@valor36az]

Great video

[@kristakingmath]

Thanks!

[@hoangtran95]

What rule do doctors follow?

[@PeterFekete]

Can we really use LH on a limit sin(x)/x? I found sources arguing that you can use LH only if the 2nd derivative exists, which means that this cant be applied on this limit, right? Isn't better just use the mclaurin expansion for sin x, get rid of the ratio and then plug the the x=0? Like (x - x^3/3! + x^5/5! - ...)/x = 1 - x^2/3! ... and after by plugging x=0, we get just 1.

[@WestonMurdock]

So, if either the numerator or denominator is equal to 0, but not both, so 0/x or x/0, is the limit just 0 so that we don't have to do anything else?

[@rajendramisir3530]

lim x-2/x = 2-2/2=0/2=0. x->2 In the first case, the limit is 0. lim x/x-2 = 2/2-2=2/0 x->2 In the second case, the function f(x)/g(x) is undefined at x=2. The function may be discontinuous at x=2.

[@quiramquimod5467]

Thanks

[@franchello1105]

Can you explain why this works? If we take a different point, let say 3, which doesn't have an indeterminate form, you get different answers (lhopital and substitution). (1/108 vs 1/65)

[@alexiossurya3868]

U may be making some Mistake

[@alexiossurya3868]

Becoz Rule is a Rule It Satisfies all conditions upon what it works ( ˘ ³˘)

[@muralithatholu2113]

my question is .....why we have derivate..what is the heart of this L'Hospital's theorem.please explain..

[@bobh6728]

The first condition to use the rule is that g’(a) 0. Then how would you ever use the second derivatives which is said can be used if the first derivatives give you 0/0 which is saying g’(a)= 0. What am I missing?

[@enickasmith4223]

Very very good video

[@kristakingmath]

Thanks Enicka!

[@ILikeReadingTho]

I wonder why all I knew about solving the problems with limits were L'Hospitals rule.

[@HarpalSINGH-df6pu]

vich hospital is dis in relation 2? i done understood ur meths

[@fishertech]

Doesnt the sin x function oscillate? then how come you still managed to use Lhopital rule?

[@fishertech]

talking about the example at 4:16

[@webtube4348]

Yes, it's true that sin x oscillates, however we are now approaching a finite number, not infinity or negative infinity, therefore L'Hopital's (Might spelled it wrong) Rule can still be applied.

[@aggravatedbaguetteshorts6503]

For the last one, you could actually simplify it and solve the derivative without using L'Hopital's rule. The denominator is a difference of perfect squares so you can rewrite it as: (x^2+4)(x^2-4). Since you have another difference of perfect squares, you can rewrite it again, all of the denominator: (x^2+4)(x+2)(x-2). Since, (x-2) is in the numerator, you can rewrite the fraction, (not going to bother trying to write limit with proper form in a KZbin comment), as 1/(x^2+4)(x+2), or 1/x^3+2x^2+4x+8. When you plug 2 in, you get that 1/(2^2+4)(2+2)=1/(8x4)=1/32. Not trying to criticize, just pointing out and helping others who would like to solve it in a different manner. :)

[@fetauAdu]

Yes, I was wondering why she didn't use the factoring too.

[@aggravatedbaguetteshorts6503]

@@fetauAdu ya it's useful

[@AyushBhattfe]

its funny that in India we say EL-Hospital rule

[@alexiossurya3868]

🤣

[@mahmoudashraf117]

already Bernoulli's theorem is enough :)

[@xOxAdnanxOx]

هههههههههههههههه hahahah this is the translation of the laugh in Arabic

[@saiphislam9396]

Imagine applying bernoullis rule again and again upto f deca prime x.

[@SKAOG21]

I'm quite sure you can't use L'Hôpital's rule for sinx/x when X approaches 0 because of differentiation from first principles, where you come across sinh/h as h approaches 0.

[@mohammedinamulhaque116]

We can apply

[@tusharsingh-gm8kz]

Good ❤️

[@kristakingmath]

Thanks! :)

[@reggienalupa5166]

how about non zero over 0, l hospital is not applicable ?

[@kristakingmath]

No. That usually means you have a vertical asymptote at that point. :)

[@reggienalupa5166]

Thanks :D your tutorial videos help me a lot

[@anvu1854]

Really sad frtim

[@johntolkien9629]

At 05:49 you use a square symbol - why ? Am reading this as 'Then g prime of a cannot be equal to zero' - but don't want to make assumptions.

[@anuvette]

It's there to check the parameter for L'hospital's rule to work

[@thebluebeyond2329]

Ah you have a KZbin channel as well 🤞😎✨✨. I'm saved. I tried taking the Precalculus course but my pockets are not that deep as dollars weigh up than a Rand

[@kristakingmath]

Glad you found me here, Kendra! :)

[@RafaelRabinovich]

I think that "s" is mute

[@hippy518]

AH ?

[@vanguard7674]

L-HOP-PITAL?

[@kristakingmath]

L'Hospital's or L'Hopital's?? You say 🍅 I say 🍅. ;)

[@gatsuofmontecristo7337]

what about eggplants?

[@tuna3875]

I think its L'Hopital, but its pronounced "Loh-Peetal" because the H is silent in French.

[@Phoebus82]

When I study maths sometimes I need a n'hospital :D

[@matth8200]

I say "tomato," you say "tosmato." :)

[@stumbling]

Eou sai tomateux, Ighe saelles teumeiteaut.

[@Davio88]

Le Hospital.

[@xdxd-br5jv]

El hospital

[@davidgruen7423]

The hospital rule if you translate it word by word

[@adityamaurya9437]

From India

[@kristakingmath]

Hi! :)

[@meiwinspoi5080]

just saw this by chance. sorry madam you cant use l’hopitals rule for sin x / x because the fundamental evaluation of derivative of sin x, you need to prove lim x tends to 0 of sin x /x = 1. so it will amount to circular argument if l’hopitals rule is used for sin x/x. you have to use epsilon delta definition of limit in this case.

[@xavierplatiau4635]

There is a little mistake in the name which made me laugh a bit as a French guy, « Françoise » is a feminine name and « marquise » also refers to a feminine person. His name is Guillaume François Antoine, marquis de L'Hôpital, sometimes written L’Hospital. By the way, hôpital is French for..an hospital. That’s why you shouldn’t pronounce the « s » in his name.

[@allanhenriques2694]

who else came here because of the thumbnail?

[@noobgamerreal234]

hospital's rule 💀

[@liam8904]

Wait a minute, she's using the pronunciation to distract us from the real problems!

[@Rogeriopris]

99% of people that explain math start from saying believe in what I am sayng here and then try to understand this magic and do the exercises as the masters teaches you.Bad explaination

[@zeroday7878]

loh-pee-tahl?

[@merc340sr]

In mathematics, more specifically calculus, L'Hôpital's rule or L'Hospital's rule (French: [lopital], English: /ˌloʊpiːˈtɑːl/, loh-pee-TAHL) provides a technique to evaluate limits of indeterminate forms. "Hospital" is the old French spelling. In this case the "S" is silent. In modern French, it's more common to use "L'Hôpital", as in de L'Hôpital's rule. source: Wikipedia. Use the latter expression and you will sound more educated....loll...