1st semester calculus was easy for me but i got a new teacher for 2nd semester calculus and i've been struggling. i just found your youtube and your videos have helped so much! thank you!!
I have been cramming for my final and without your help I would have spent many frustrating hours. It is 3 am and I have to get up for my exam in three hours. I am feeling so much more confident thanks to you. THANK YOU!
@IGNsucks bernoulli rocks (which one though). those poor brothers fought. one day, when you all get rich, you can pay me to do some math research but you all get all the credit.
@patrickJMT John or Johan, I think. the brother that died was Jacob so the other one. Either way cant feel sorry for the guy considering he even fought with his son, what a terrible father. and I dont think he payed for them (technically) I know Lhopital was his student and that Bernoulli let him use his findings and that Lhopital published those notes. either way its stealing but again Bernoulli was an ass so who cares.
@patrickJMT yeah im doin a research thing on him right now for my ap calc class, he studied under johann bernoulli and had his own findings as well. when this method was learned by others, it was believed that l'hopital had come up with it which was not l'hopital's intention, bernoulli was only a little upset with this. i guess there's reason to believe he either came up with it himself, or bernoulli did, or they both found it together lol, but later l'hopital had paid bernoulli off for credit
@AirJordanXVIII I know this is four months old, but the reason that it's spelled in two different ways is because the french word Hospital was changed to Hôpital for faster pronounciation. Most words with an S that jus slowed down the pronounciation of the word had their S removed and a "circumflex" accent was added to the vowel closest to the S.
i) "Exponentials Win" As x goes to infinity, e^x dominates all polynomials, so the answer should be immediately obvious (though of course knowing the answer, and proving it, aren't the same thing). As x goes to infinity, it's essentially (e^x)^(1/x), which is e.
ii) In terms of L'Hospital's Rule, that e^x dominates all polynomials shows up because, once it's in a form where L'Hospital applies, taking derivatives never removes the exponentials, but does drop the degree of the polynomial.
iii) It's solvable algebraically, assuming (as calc students should know, or L'Hopital) that x / (e^x) goes to 0 as x goes to infinity. Factor out e^x and it's e * (1 + x / (e^x) )^(1/x). The 2nd factor is no longer indeterminate (it's form is (1+0)^0, so goes to 1).
Ex 2:
i) Look for simplifying transformations. Substitute u = 3x, and it becomes ( cos(u) ^ (1/u) ) ^ 15, so the problem is really cos(u) ^ (1/u) as u (=3x) goes to 0. No more 3's and 5's will appear in the calculations.
ii) Whenever there's a function raised to a function, it's good to think about the domains for a moment. In general, since the power will only rarely hit an integer or odd-denominatored rational number, x's making the base function negative will usually be excluded from the domain. Also, x's making the base function is 0 will also be excluded if the power there is negative. Since cos(3x) is near 1 when x is near 0, there's no problem here.
iii) You can approximate cos(u) using the degree 2 Taylor polynomial at u=0, then make a substitution z = (u^2)/2. Have z goes to 0+ and u = sqrt(2z) (loses the neg u's). It will then look like (YouTube isn't letting put in the steps) (1/e)^0 = 1. Thus expect cos(u)^(1/u) goes to 1 as u goes to 0 (so 1^15=1=answer). This isn't rigorous (L'Hospital is!) as it uses an order 3 approx for cos(u). It's just showing another way that's sometimes useful.
Hey, thanks for the vid, why do we have to take the lny on the left if we dont end up taking the derrivative of lny, can we just ignore the left side or lny?
Why is 1 to power of infinity considered indeterminate. Intuitively, I would have thought that 1 to the power of anything should be 1, regardless of how many times you raise it to a power. Can someone give an example of a limit problem where 1 to the power infinity doesn't give 1...
as a number gets diveded by numbers betwin 0 and 1 they get larger as the numbers they are devided by get closer to 0. and so closer to infinite as x->0 we say 5/x-> infinite. I hope this is helpfull
Im in the uk, first year University, Im curious, is this stuff what kids are taught in high school in america? im seeing this stuff only just now.... :/
Ln is a function, not a number, and so you can't multiply the brackets out. We are taking the natural log of their sum and we cannot split them up.
Also, we can't cancel (x + ln/x)/x. If we say 1*1E999...(etc) equals infinity then 2*1E999... equals infinty. So 2*1E999/1*1E999 = infinty over infinty = 1. But 2*1E999/1*1E999 = 2. So we can only cancel infinty when we know it is exactly the same. In the example - we know e^x = e^x and so we can cancel.
I understand what you're saying, but I hate to burst your bubble, in math, it.s all about solving the problem in the most effective way( that's usually the shortest way). Trying to solve it like you do is good practice for trying to figure out how to solve things in general, but it's highly impractical. when you have to solve a lot of stuff in 1 h , like i used to have to in high school. and still do in college, L'H is very useful. as I said, your way is not wrong, just very impractical.
I think it's ridiculous not to use L'Hopital when you can. It helped me so much when dealing with limits. And with L'H you know you're not wrong, when with trying like you do might lead you to a false result. I love L'hopital too. Why should you try to complicate things even more?
I'm learning it in high school and i never thought of it that way! I guess i'm grateful that my teacher explains it step by step but it's still AP so they go really fast. Thankfully there are videos like this one. Thank you Patrick!
b/c the standards are different in college or uni. i know its messed up but they dont teach it like high school step by step with patience they assume u are "intelligent enough to get them right away and skip some steps in ur head like they do "
wtf? why does my professor getting 90$k and still do not how to teach stuff in these simple way........by the way my professor was moron ....do not know how the overhead works......thanks...... it will be great if i have find it earlier.....
Thank you so much! You're great at explaining. I am a little confused. At 6:17, why is it raised to an "infinity" when 5/x? 5/0 is undefined though.
colors0901 1 week ago
Your work is so freaking beautiful,Thank God for creating you haha
Gixxiemoto 2 weeks ago
This was very helpful.
Shkencetari 1 month ago
i am so feeling this guy..just 20 mins has helped me alot..hehe.thanks men.
1603stanley 2 months ago
I thought (1/cos(3x))=sec(3x) not tan(3x).
maccamracecar 2 months ago
@maccamracecar nvm I see my fault.
maccamracecar 2 months ago
Why did you write y as e^ln(y)?
sfrancisco18 3 months ago
On the second example why does the 5 stay for the ride and not turn to 0 when L'Hospital's rule is applied?
tifrogers44 4 months ago
@tifrogers44 5 is a constant, when you take a limit of a constant, it equals the constant
mistersirisaacnewton 3 months ago
I thought it was spelled L'Hopital's Rule o.o
Angelgrrl04 4 months ago
@Angelgrrl04 My professor said it has many spellings.
maccamracecar 2 months ago
Why is 1^infinity indeterminate ? I thought that it would just be 1x1x1....=1. Can you explain that please ? :) Thanks. :D
AceAites 4 months ago
@AceAites cause it is a limit... the value is getting close to 1 , but not necessarily equal.
patrickJMT 4 months ago
@AceAites so when you raise that number to a large power, many things can happen.
patrickJMT 4 months ago
@patrickJMT is that just for 1^infinity or does any number to the power of infinity become indeterminate?
applesnnbananas 4 months ago
@applesnnbananas 1^infinity only.
patrickJMT 4 months ago
left handers!
LB2102 6 months ago
1. lim→0 x^sin(3x)
2 f(x)= (e^x^2-1)/x
sharbini89 8 months ago
i have to learn all of this by tomorrow :(
ezalcinoryahoocom 8 months ago
What a legend!
zinhle23 9 months ago
thanks a lot! i didn't understand this when it was taught to us today. you always save me from math.
teethstainedred 9 months ago
uhmm ThanK You very much 4 this.. but y do we use e^(0)? i mean y use e??
SBUDEX 10 months ago
awesomastic.
MultiMoe91 11 months ago
love maths!!
Quacyify 11 months ago
man......... thanks alot.............. it helped me... coz ur examples are the same as my teacher's example which i didn't understand.........
it really amazed me the way u teach..
:)
chiconium 11 months ago
1st semester calculus was easy for me but i got a new teacher for 2nd semester calculus and i've been struggling. i just found your youtube and your videos have helped so much! thank you!!
mtech1020 1 year ago
thank u! i was so lost on this stuff. hopefully i'll pass that quiz tmrw
BlahLab 1 year ago
Brilliant method of teaching, but it's actually pronounced "la ho pital"
BigOlBagOfWhatsNew 1 year ago
Can someone tell me why 5/0 is equal to infinity? Normally, it's undefined... so is infinity in some cases the same thing as undefined?
akmcferran 1 year ago
@akmcferran its supposed to be as it x approaches 0 then as x gets smaller 5/x goes to infinity
skinnycarrotchan 11 months ago
doesn't differentiation of 1/x give -1/x^2 ?
please explain the first differentiation part again
howcome you got one in the bottom?
lifesabigfightbaby 1 year ago
funny how i pay for school and my instructor never discussed this.
thanks
chr0nicForc3 1 year ago
OMG!Why can't I have a brain like yours!!!
lady000butterfly 1 year ago
I have been cramming for my final and without your help I would have spent many frustrating hours. It is 3 am and I have to get up for my exam in three hours. I am feeling so much more confident thanks to you. THANK YOU!
ellaweim 1 year ago
ur a boss ;)
ewd199 1 year ago
ur a boss ;)
ewd199 1 year ago
ur a boss;)
ewd199 1 year ago
don't you have to use the power rule when you're taking the derivative in the second problem. the (5/x)(ln(cos3x) ?
FallenOracleX 1 year ago
I wonder why I can't understand it when my asshole professor tries to teach it to me but I suddenly can when this guy does it
kamikakushii 1 year ago
THank youUU!! ExtREmeLY HelpFuLL:)
111000may 1 year ago 2
it's pronounced lawn for ln! do you spell out log or do you just say log?
spectralblue 1 year ago
good job. thank you.
may you be blessed to make many more awesomely helpful videos
geekgal1000 1 year ago
Your a great teacher!!! =D
x0chick 1 year ago
seriosuly you should be a teacher. btw its l'hopitals, not l'hospital haha. but your great man tahnks a ton
AirJordanXVIII 1 year ago
@AirJordanXVIII actually, both spellings are common
patrickJMT 1 year ago 8
@AirJordanXVIII it doesnt even matter the guy stole the idea anyways, it shouldnt even be called l'hopitals rule.
IGNsucks 1 year ago
@IGNsucks bernoulli rocks (which one though). those poor brothers fought. one day, when you all get rich, you can pay me to do some math research but you all get all the credit.
patrickJMT 1 year ago 5
@patrickJMT John or Johan, I think. the brother that died was Jacob so the other one. Either way cant feel sorry for the guy considering he even fought with his son, what a terrible father. and I dont think he payed for them (technically) I know Lhopital was his student and that Bernoulli let him use his findings and that Lhopital published those notes. either way its stealing but again Bernoulli was an ass so who cares.
IGNsucks 1 year ago
@IGNsucks and he really did not steal it, he paid for it!
patrickJMT 1 year ago 3
@patrickJMT yeah im doin a research thing on him right now for my ap calc class, he studied under johann bernoulli and had his own findings as well. when this method was learned by others, it was believed that l'hopital had come up with it which was not l'hopital's intention, bernoulli was only a little upset with this. i guess there's reason to believe he either came up with it himself, or bernoulli did, or they both found it together lol, but later l'hopital had paid bernoulli off for credit
dmd70621 9 months ago
@AirJordanXVIII Lol. The Hospital's Rule.
Patindahat07 1 year ago
@AirJordanXVIII I know this is four months old, but the reason that it's spelled in two different ways is because the french word Hospital was changed to Hôpital for faster pronounciation. Most words with an S that jus slowed down the pronounciation of the word had their S removed and a "circumflex" accent was added to the vowel closest to the S.
Hospital ----> L'Hospital
Hôpital-----> L'Hôpital
Hostel
Hôtel
Arolbreaker 1 year ago
Nice vid. The steps are well spaced, clean, and clear. I have some non-L'Hospital comments about the particular problems.
VeryEvilPettingZoo 1 year ago
Ex 1:
i) "Exponentials Win" As x goes to infinity, e^x dominates all polynomials, so the answer should be immediately obvious (though of course knowing the answer, and proving it, aren't the same thing). As x goes to infinity, it's essentially (e^x)^(1/x), which is e.
ii) In terms of L'Hospital's Rule, that e^x dominates all polynomials shows up because, once it's in a form where L'Hospital applies, taking derivatives never removes the exponentials, but does drop the degree of the polynomial.
VeryEvilPettingZoo 1 year ago
(cont)
iii) It's solvable algebraically, assuming (as calc students should know, or L'Hopital) that x / (e^x) goes to 0 as x goes to infinity. Factor out e^x and it's e * (1 + x / (e^x) )^(1/x). The 2nd factor is no longer indeterminate (it's form is (1+0)^0, so goes to 1).
Ex 2:
i) Look for simplifying transformations. Substitute u = 3x, and it becomes ( cos(u) ^ (1/u) ) ^ 15, so the problem is really cos(u) ^ (1/u) as u (=3x) goes to 0. No more 3's and 5's will appear in the calculations.
VeryEvilPettingZoo 1 year ago
(cont)
ii) Whenever there's a function raised to a function, it's good to think about the domains for a moment. In general, since the power will only rarely hit an integer or odd-denominatored rational number, x's making the base function negative will usually be excluded from the domain. Also, x's making the base function is 0 will also be excluded if the power there is negative. Since cos(3x) is near 1 when x is near 0, there's no problem here.
VeryEvilPettingZoo 1 year ago
(cont)
iii) You can approximate cos(u) using the degree 2 Taylor polynomial at u=0, then make a substitution z = (u^2)/2. Have z goes to 0+ and u = sqrt(2z) (loses the neg u's). It will then look like (YouTube isn't letting put in the steps) (1/e)^0 = 1. Thus expect cos(u)^(1/u) goes to 1 as u goes to 0 (so 1^15=1=answer). This isn't rigorous (L'Hospital is!) as it uses an order 3 approx for cos(u). It's just showing another way that's sometimes useful.
VeryEvilPettingZoo 1 year ago
Thanks, very helpful, your a great teacher.
yoguely 1 year ago
@yoguely thanks!
patrickJMT 1 year ago
This has been flagged as spam show
Hey, thanks for the vid, why do we have to take the lny on the left if we dont end up taking the derrivative of lny, can we just ignore the left side or lny?
lexyparis 1 year ago
Hey thanks for the vid, I was wondering why you take the lny on the left and then not take the derrivative of it, can we just ignore the left side?
lexyparis 1 year ago
Hey Patrick your teaching really helped me out for my exam.. cheers
thomofmx 1 year ago
Why is 1 to power of infinity considered indeterminate. Intuitively, I would have thought that 1 to the power of anything should be 1, regardless of how many times you raise it to a power. Can someone give an example of a limit problem where 1 to the power infinity doesn't give 1...
Thanks
hifhif123 1 year ago
@hifhif123 take for example the limit of: (1+1/n)^n as n goes to infinity.
this limit is a case of 1 raised to infinity,but the answer is actually the number e.
this is one of the various definitions of e.
dvary89 1 year ago
@dvary89 Sweet! Can't believe I didn't think of the definition for e.
hifhif123 1 year ago
hey thanks pattric u er really very great in teaching,this vidio of urs is like a magic wand which make solving limits easy.thanks a lot
supermadcrazy 1 year ago
@supermadcrazy haahahah! glad i can help
patrickJMT 1 year ago
great videos.thanks
pyarexxx 1 year ago
thanks for the video, this way i didnt have to search through my notes to way back in the year
jkau94 1 year ago
great video keep up the good work :D
jparra140 1 year ago
brain fart and this just helped a lot,THANK YOU!
mikeybabies 1 year ago
i actually have atest on this tomorrow and i'm watching this vid to help me haha. thx so much!!! it's a great refresher and helper
paradiseheaven 1 year ago
u r greaaaaaaat!!
thnx =]
7kholoud 1 year ago
THANK YOU SO MUCH. you're better than my calc "prof" at explaining the material. just thought i should let you know
colmoses1337 1 year ago
You're a lifesaver. I've used your videos a lot. A lot a lot. Many thanks, sir.
ldailyfan 1 year ago
Patrick you are the man!
flaco954 1 year ago
Is this the way for all tricky questions?
Amato1Allah 2 years ago
Thanks, that was a great explaination
sethyup 2 years ago
Great explanations, perfect in case my APC kids are tired of listening to me =] thanks!
mgironda 2 years ago
Hi at 7:30, shouldn't it be +15, not -15 ?
Thanks
shoaib21soccer 2 years ago
@shoaib21soccer
He took the negative from the Sin, so that way he had -15, so that it was just sin/cos to give him tangent. rather than -sin.
Epeetastic 2 years ago
o ight thankssss
shoaib21soccer 2 years ago
Thanks from Canada, eh!
CompsciWiz 2 years ago
YOU ARE GREAT
AlaaJassar 2 years ago
You're helping people from all over the world. Thank you from Argentina.
sebastian123sanchez 2 years ago
thank you from canada
alternativepirate 2 years ago
thank you from Viet Nam
lawliet110 2 years ago
Thanks from Barcelona
geri71 2 years ago 2
love Barcelona!
goaram 2 years ago 2
hats off
Yaronthegreat 2 years ago 3
Man, your videos are awesome. I can't even begin to tell you how much they've helped me out.
ElmoPansy 2 years ago 10
u r just straight up amazing! if i get through my college Cal 1 class because of u i'm gonna donate to ur website
dozentcare 2 years ago 11
i thought 1/0 is undefined. How then does 5/0 equal to infinity?
freemkulima 2 years ago
as a number gets diveded by numbers betwin 0 and 1 they get larger as the numbers they are devided by get closer to 0. and so closer to infinite as x->0 we say 5/x-> infinite. I hope this is helpfull
gigijaponezu 2 years ago
la verdad q muy bien explicado.y eso q esta en ingles je je je pero las matematicas no tienen idioma juaz!!!baez!
doncprinc 2 years ago
I don't speak spanish extremely well, but a rough translation is:
The truth is that the explanation was very well. And it is in english...???...but math doesn't have a set language!
Did I translate that right? :O
shogundapker 2 years ago 3
sorry i meant when ln|infinity| it's 0 ....nt 1
RadoAller 2 years ago
Instead of using L'Hopital's Rule could i say that becuz it's ln|infinity| so it's 1....cuz that's when ln is infinity?????
RadoAller 2 years ago
Im in the uk, first year University, Im curious, is this stuff what kids are taught in high school in america? im seeing this stuff only just now.... :/
beep1234beeper 2 years ago
nop,,,we take it in uni too.......:P
RadoAller 2 years ago
Well, this is what I learned in high school, at least for the last two years of it.
Arycke 2 years ago
This comment has received too many negative votes show
only the smart asians take calculus in high school in America.
itsthebanshee 2 years ago
ahhh, nothing like some good ole fashion racism.
patrickJMT 2 years ago 6
Not Racism, but racial stereotyping
Aaradmegaman234 2 years ago
@patrickJMT
haha, what a hater!! math is for everyone :D
Slayermagic 1 year ago
lol
Haroson 2 years ago
Thank you so much, this is really clear and helpful.
kelsey1131 2 years ago
i worked it out slightly differently but I'm not sure if its right but here it is, after i reached
ln(e^x + x) / x
i worked out the brackets and got:
lne^x + lnx / x
i then used the property of logs and brought down the x in lne^x:
xlne + lnx / x
since we know that lne=1, we get:
LIM x - infinity = x + lnx / x = infinity over infinity
chriskill08 3 years ago
Unfortunately it doesn't work like that.
Ln is a function, not a number, and so you can't multiply the brackets out. We are taking the natural log of their sum and we cannot split them up.
Also, we can't cancel (x + ln/x)/x. If we say 1*1E999...(etc) equals infinity then 2*1E999... equals infinty. So 2*1E999/1*1E999 = infinty over infinty = 1. But 2*1E999/1*1E999 = 2. So we can only cancel infinty when we know it is exactly the same. In the example - we know e^x = e^x and so we can cancel.
BigDaNNyBoi1989 2 years ago
thanks man...i guess the answer i got was a coincidence then
chriskill08 2 years ago
we now use l'hopital rule and get:
1 + 1/x / 1 = 1 + 1/x (only one "1" is over x)
i then worked out the fraction and get:
x + 1 / x ( both x and 1 is over x )
this still gives infinity over infinity so i use the rule again
1 / 1 = 1
therefore i still get the answer but i'm not sure if its right. can you please check Patrick
chriskill08 3 years ago
Once again, great demonstration, PJMT.
Syruscleat 3 years ago 2
thank ya mr. syrus
patrickJMT 3 years ago
I understand what you're saying, but I hate to burst your bubble, in math, it.s all about solving the problem in the most effective way( that's usually the shortest way). Trying to solve it like you do is good practice for trying to figure out how to solve things in general, but it's highly impractical. when you have to solve a lot of stuff in 1 h , like i used to have to in high school. and still do in college, L'H is very useful. as I said, your way is not wrong, just very impractical.
sakuranohana89 3 years ago 2
I think it's ridiculous not to use L'Hopital when you can. It helped me so much when dealing with limits. And with L'H you know you're not wrong, when with trying like you do might lead you to a false result. I love L'hopital too. Why should you try to complicate things even more?
sakuranohana89 3 years ago
i love l'hospital!
patrickJMT 3 years ago
whoa that's crazy!
what do you guys do instead of L'hop??
dapoochiman 3 years ago
:D I'm doing this right now in Calculus II.
zomgmoose 3 years ago
I did that long time ago on Calculus 1...
modem09 3 years ago
i did that in calculus 0
patrickJMT 3 years ago
I'm learning it in high school and i never thought of it that way! I guess i'm grateful that my teacher explains it step by step but it's still AP so they go really fast. Thankfully there are videos like this one. Thank you Patrick!
cottonkandynb 3 years ago 2
b/c the standards are different in college or uni. i know its messed up but they dont teach it like high school step by step with patience they assume u are "intelligent enough to get them right away and skip some steps in ur head like they do "
:-s
joojoo00oo 3 years ago
wtf? why does my professor getting 90$k and still do not how to teach stuff in these simple way........by the way my professor was moron ....do not know how the overhead works......thanks...... it will be great if i have find it earlier.....
manoj0072006 3 years ago
i am your teacher now! send me your tuition! i will print out a nice diploma from me!
patrickJMT 3 years ago