JEE Advanced 2022 #2
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- Опубликовано: 2 окт 2024
- This this limit problem is the limit of a composition of trig and logarithmic function. The main idea is to know how to compose the functions and take the limit of the composed function as the function of the limit.
Cool!❤❤Hey one more amazing question from J.Adv
Hey prime newton, i am a jee aspirant for 2026 jee advanced and also a big fan of yours...
Can you pin me plz ❤
hey could you please clear my doubt if i had done the question the right way or not so here's what i did, so basically limit means that x is approaching alpha from the positive so root x will approximately be qual to root alpha, therefor the the equation will become log with base e and no. 0 so its value will be 1. now the equation will be simplified to sin(2pi/12) which is equal to 1/2
L'Hospital's rule was needed to apply only twice. You could have separated the exponential from the denominator after applying the rule first and before applying it second time.
I know it's not rigorous or even right but i just saw that as x→a+
ln(✓x - ✓a)→ln(0) and ln(e^✓x-e✓a)→ln(0) so injust cancelled them out and got as x→a+ g(x)=2 and plugging 2 in f(x) we get ½. It's not at all the right way but funny how i got it.
Much simpler and faster resolution (without any l’Hôpital’s rule) by use instead of simple rearrangements :
Let X=sqrt(x) and A = sqrt(alpha), thus :
ln{expX-expA} = ln(X-A) + ln{[expX-expA]/(X-A)} = ln(X-A) + W
And : g(x) = 2 ln(X-A)/ln{expX-expA} = 2/[1+W/ln(X-A)]
Thus it’s finished !
Because W has obvious FINITE limit A when X->A (since by DEFINITION of the derivative at X=A,
Lim{[expX-expA]/(X-A)} = exp’(A) = exp(A))
Therefore the infinite ln(X-A) kills W.
Thus g(x)->2 when x->alpha
And finally, by continuity of Sinus function :
f(g(x))-> Sin(Pi/6)=1/2
0:17 thanks again for giving us your e-mail!
0:31 when i watch your videos i become more and more sure with my thought that any mathamatical problem is solvable
and by the way sry for my late commenting!
1:52 and now even put in sth. for the x there and simplify
2:12 yes sorry that is that what i mean
5:08 i thought you have mention that before haven´t you?
6:36 ok "derive" i learned a new vocabulary xD
7:48 but please move the tops and botems of the big dividetation and simpliphy
8:03 ok to casel out before make sence yes ok
10:04 ok the only part i realy understand :D
10:41 finaly!
12:00 how ever nice video again!
Excellent explanation Sir. Thanks 🙏🙏🙏🙏🙏🙏
The second use of l’Hôpital’s rule is obviously unnecessary since the limit is simply the DEFINITION of the derivative of the exponential function at A=sqrt(alpha) with X=sqrt(x) :
Lim{(expX-expA)/(X-A)} = exp’(A) = exp(A)
Thus by PRODUCT of two continuous functions :
Lim{(expX-expA)/[exp(X)(X-A)]} = Lim{(expX-expA)/(X-A)} / Lim{expX} = exp(A)/exp(A) = 1.
QED
My comment might seem strange, but do you have a video on the rules for setting parentheses. Where I live we use different conventions about that. Say we never out a single positive (unsigned number in the parenthesis) usually we put parenthesis when we have two action signs meet, e.g. -(-...) , +(-...) or *(-...). Or when the order of operations could be ambiguous like in a famous meme where different calculators give different answers, something like 6/3*2.
From that point of view the right upper line at the beginning of the video appears ambiguous to me. If I saw that apart from the video I would perceive it as sin(π/12) multiplied by some factor, I wouldn't guess it's the part of the sine's argument.
So , do you have some vudeo maybe short about that. I'm sure it could be useful for many,even those who have studied it at school and especially those who studied in another types of school system. Thanks 🙏 I find your videos very interesting and useful
Blessed be Gauss, Mat is the Queen of ciences....
Thanks for an other video master
8:57 From here you could separate this into two limits by product rule of limits that is,
(lim x-> alpha+ of 1/e^sqrt(x))×(lim x-> alpha+ of ((e^sqrt(x)-e^sqrt(alpha))/(sqrt(x)-sqrt(alpha))) Now the first limit in that product is easy to compute as it is just 1/e^sqrt(alpha) the second one in the product can be identified as the limit definition of the derivative, namely the derivative of e^sqrt(alpha) with respect to the variable sqrt(alpha) (note: we are not differntiating wrt alpha) which is exactly e^sqrt(alpha) and now the limit has become the product of two reciprocals which is just 1 and so the limit is one.
I solved it by series expansion I didn't use l hospital rule
Was it faster?
Yea I too think it is more convenient@@PrimeNewtons
@@PrimeNewtonssir first of all thanks for replying me I learnt a lot from u , I am a jee aspirant and I am in class 10. Sir I am currently focusing on my mathematical manipulation that's why I don't use l hospital rule cause it make the solution very simple and easy. That is why I gave it a try by using series expansion and I used the information that alpha is positive. Very very thanks to you I attempt all the questions you share to us . Again 🙏🙏
Am also in 10th how much syllabus have u covered of jee@@VineetaSingh-m5j
@@VineetaSingh-m5jho𝘀pital
For someone who has a major in culinary arts and a minor in math, you have an astounding understanding of mathematics and an astounding ability to explain it very well done
Honestly thats quite a weird combination, major in culinary arts and minor in maths.. why did you take that combination, if i may ask?
i literally tried this yesterday in my book and youtube recommended me the same question lmao what are the odds
Sir reading in class XII and this year will appear in JEE Mains and I have solved it in n 10 second
❤
God has certainly gifted you as a teacher. Thank you so much for these videos.
2 times L hospital, only ....
We can insert the limit because sin (outside function) is continuous on R???
Thanks for an other video master
Was gonna say u need a bracket on ur first line.
I think since Sin is a periodic function, it will have the same value of pi/6 for every naturnal number multiple of pi/6, or I might be wrong
2pi / 6 is also a natural multiple of pi/6 but it equals pi/3. Then the Sin of it would be root 3 / 2
The correct way to use the periodic nature of sin (or any trig function) is that adding 2 pi doesn't change its value. Other conclusions may not be correct
Also IITs are one of the top most colleges in the world. The only reason they don’t have their name on top in the list where IVY colleges are at top is because they have set some different kind of standards which are against most Indian and other Asian institutes
Just imaging how to resolve lim f(x)= lim f(g(x)) !
5:57 why does the derivative of sqrt alpha go to 0?
@@SpandanJawkhedkar lol I realized that after rewinding.
@@Th3OneWhoWaitscause alpha is a constant in question
Brialliant ... !
Great..!,Sir...Thank You...