Lecture 04: Continuity of multivariable functions

[@rudraksh5840]

€ and ∆ scared the shit outta me in college. Good books help through such bottle necks. Confused students can consult some basic topics related to real analysis (especially about proofs).

[@himanshusharma9981]

which books??

[@foofoogaming551]

it really helped me but i am still confused in putting the inequality signs

[@naveengupta9680]

Exactly my problem

[@MasterOwen69]

Yeah same problem here

[@Raj-bu7kg]

In the last limit... Lim (x tends to 0) Sin(1/x) will have value from (-1,1) and when Lim(y tends to 0) then value will become 0 of whole limit. Let me know where I am wrong. Your lectures are helpful :)

[@RahulYadav-es4dn]

Hi Raj, Lim(x tends to 0) sin(1/x) is not defined as 1/x is not defined as x=0. I know that you might think, that the value of sin always lies between [-1, 1] but here the angle itself is not defined then how come the value of sin will be defined.

[@priyanshkumar17]

For the second iterated limit , y is treated constant and is thrown out of the inner limit. And we know from our JEE days , lim.(x tends to 0) sin(1/x) is non-existant.

[@parmitha5807]

hey Raj, I think you are exactly right

[@madhutammisetty5597]

Sir!!! you gave a great clarity thanq

[@ojasmarghade3694]

what a amazing lecture sir

[@nitishbudhiraja7687]

can we put y equals to mx^2 in the second example you have covered in this video. by putting this we will get 1/1+m^2, so can we directly say limit do not exists from this

[@shikhargovil9579]

yes we can do it.

[@lavanyapal9912]

Thank You Sir, you are legend.

[@AnkitKumar-cr3qs]

9:00 sir why did you Show the limit of f(x,y) is zero..why didn't you put y=mx.^2??..when I put y=mx.^2 then m doesn't divide and m remain in inequality so f(x,y) depends on m this implies the limit of function doesn't exit....but apart from it when we use epsilon Delta epsilon then we get Delta and function will exist That's why sir some confusion between epsilon-delta definition between y=mx.^2🤔

[@deepakkhandelwal3463]

m didn't cancel out but so does x 3 power in numerator and 2 power in denominator for x So lim x to 0 makes the whole expression 0

[@vikashnigam2115]

@@deepakkhandelwal3463nice brother

[@Ashifa_shaikh]

😂sir, some of ur lectures are too much good but in few of them u don't explain much .but i can understand student of iits r able to understand all that stuff behind ur given clue😂btw thanks .♠️

[@vikas7979]

😂😂

[@prathams7432]

Exactly😂

[@jaypatel7632]

Very great😁🎉👍...

[@suryansh8066]

AT 11:10 ISINT IT SHOULD BE MINUS INSTEAD OF PLUS??

[@siddharth4074]

Wahi to waha answer 0 araha hai Limit doesnot exist ayega

[@suryansh8066]

No I was wrong due to mod the negative changes to positive the given answer is right 👍

[@gyanaranjanpanda3925]

sir at 12.42 ofthis video you have expandede and came to a result that expression is =mod(x^2-y^2) and solve??

[@Batman00951]

Same doubt

[@vedangpathak]

Yes you can do that, and it would be more correct as delta equal to epsilon over 2 will be biggest possible delta such that all the points in the delta neighborhood gives output within epsilon nbd.

[@AnshKumar-hm8ts]

i have a question that when one iterated limit does not exists then how can we say that the function is continous at (0,0)?

[@panashetengenyika789]

Thank you sir ...but try your board is a bit far away for us to see especially when using cell phones

[@panashetengenyika789]

I mean your board is far..or enlarge your handwriting...kkkkk

[@shantanukumar4081]

REALLY HELPFUL

[@bbd9106]

khud solve kr lo sab batana mt q kiya kis liye kiya.or y to batana hi mat ki knse question me ky use hoga.great teaching

[@akashbiswas5617]

thank you sir for iterated limits make me learn so easily

[@mihirsakaria6690]

sir , in the last example it will not be continuous when y not=0.. it didn't mention to check at origin ?

[@sahilkakkar2156]

f(x,y) = 0, for all values of y, where x=0. This also includes the point (0,0)

[@AnkitKumar-cr3qs]

At 14.58 sir f(x,y)=ysin(1/x) when we put y=mx.^2 then we get f(x,y)=mx.^2sin(1/x) then function is depending on m then How can you say that function exists at 0,0 with the help of epsilon-delta definition

[@priyanshkumar17]

How the function f(x,y) is depending on m ?? When you evaluate limit x tending to zero , the limit comes independent of m , i.e. zero.

[@shreyashagrawal2264]

in the question 2x(x^2-y^2)/(x^2+y^2) if we approach the point (0,0) from the x-axis then the limit comes out to be 2x and when we approach it from the y-axis then it comes out to be 0 then how is it continuous??????

[@anujbirani3771]

But x also tends to to 0 so overall 0 from every path

[@Benzene6.023]

Suggest me which i follow during this course.

[@ajipremium2038]

Is there any rule exist like L'Hospitals rule for these multivariable function

[@highermathematics6494]

Sir, your only checking at (0,0). If f(x,y)= (1-y)/x is continous at ( 1,0)?

[@BTCIVChandanRaj]

Thank you so much sir.

[@gainknowledge285]

Sir how it drwa a graph

[@lalitsinghkarki3688]

Actually professor at 7:34 you said to move along y=0 this was I think a small error because to move along y and towards zero and also to say to put y=0 are two different things

[@pratikshamahajan3213]

thankuu...

[@MdKaif-iy5je]

But sir in ques ySin(1/x), if we take path y=mx and try to find the lim as x approaches 0 then we get it as equal to m. So the lim is path dependent and should not exist yet by δ ε method we are getting limit exist. How and why????😨😨😨😨

[@priyanshkumar17]

we get limit of mxsin(1/x) when x approaches zero to be 0 not m.

[@preetipreeti6799]

that's means if given function is homogenious then we will use like that y=mx

[@priyanshkumar17]

Yes.

[@anandakumaranujan8586]

how did you find the function is resolved by the delta turbine method or to go to the other methods

[@suryasingh9526]

coz while doing so, we have set a particular limit i.e x-del & x+del and also limit of y too. The result we get will be generalized in the whole portion.

[@ughhh5745]

Can't we solve the 2nd question with y=mx?

[@nikitapawale8115]

in the last example we use double limit concept then function is continues and when we use itreted limit function is discontinues . its little bit confusing

[@aniketiitdelhi4685]

Iterated limit is only valid iff individual limits exist, otherwise we can't take iterated limit

[@priyanshkumar17]

Iterated limit doesn't work as LHL & RHL for checking continuity of the function.

[@scholar-mj3om]

Marvelous💯💯

[@Ladki_on_fire]

Sir , inequality lagane mai problem ho raha hai

[@LapCreativeStudio]

sir its a humble request to not zoom out of whiteboard

[@iSmartRazat]

Helpful

[@vanamabhinaytinku6552]

Thank you sir

[@abhinandanmehra7765]

Sir i have a question over there that at 16:21 why you took delta equal to epsilon why it can't be less than or equal to epsilon or greater than or equal to epsilon?.

[@yadavindu]

@Abhinandan, as per the definition, for existence of limit, we just have to show that there exist a delta for any epsilon which satisfy the condition and yes it can be any delta but it has to satisfy the condition of limit,you can watch his first video on limit. Hope this helps.

[@Placement360India]

Why we are not discussing about the existence of limit when x tends to (a_+) or (a_-) as we used to do in case of single variable....

[@geetatiwari5781]

Here we are talking about functions defined from R^n to R. Refer to lecture 02.

[@MasterOwen69]

Could you please provide some reference books where all these topics are covered with good examples and exercises

[@lmaovyu]

Thomas calculus

[@anijosheyy]

Several Real Variables by S.Kantorovitz

[@Piyush-me9nu]

Thomas Calculus, BS Grewal

[@Aryan_Kashyap]

​@@Piyush-me9nu who recommended u this ? & From which clg u r

[@bhaktisagar9603]

Thank you sir.

[@integration3581]

Excellent work

[@dracomalfoy2792]

😂

[@IBvlogs]

Sir at 13:05 why is it 4|x|

[@ravichaniyara2624]

Jam k Dusare topics ka videos banao

[@RedefiningtheConcepts]

Sir I am unable to understand at which time what we to choose which path .

[@ranjityadav9190]

Great

[@AA-xk8kx]

sir how do we decide the paths? im so confused

[@idksorry3782]

Bro Have you got the answer ? If yes then tell me too

[@AA-xk8kx]

@@idksorry3782 no i havent :(

[@idksorry3782]

@@AA-xk8kx bro konse college se ho and give your IG ..I will explain

[@abhikumar4332]

Nice

[@mohamedaboustit561]

very good lecture

[@vnitian2233]

Sir graph theory ka video chaie for graduation

[@maheshwarimukesh8115]

sir yadi aap hindi me teaching karenge than your subscription is increase more

[@BabasVerse]

Sab gota ha

[@BabasVerse]

You are not able to interpret things properly

[@akashsinha9924]

Tu karde bhai

[@Maths_Magic_]

Bhai vo iit ke professor hai highly qualified understood 😎

[@shivamagrahari5425]

Sir!!! you gave a great clarity thanq