In my math class lectures at the university while studying multivariable functions the lecturer never mentioned L'hopital's rule for these multivariate functions..But in a tutorial class,a tutorial assistant approached this question..find lim (x,y)-->(0,0) [sin(x^2+y^2)]/(x^2+y^2)..by implicitly differentiating the numerator and denominator to get [xcos(x^2+y^2)+ycos(x^2+y^2)]/(x+y)...{mind you,I know,there are no indications for dx and dy,thats the approach I am just quoting it down!] then there was more implicit differentiation [the aim was as usual to remove variability of the denominator function]..the result was [cos(x^2+y^2)-2x^2sin(x^2+y^2)-4xysin(x^2+y^2)-2y^2sin(x^2+y^2)+sin(x^2+y^2)]/2...again no dx,dy or whateva!but by approach a limit lim (x,y)-->(0,0) you get 1...which is obviously the actual answer!I know this so because anyways I can the function f(x,y)=z and change the limit from lim (x,y)->(0,0) to lim z->0 so the whole thing is lim z->0 (sin z)/z which is a mathematical fact to be equal to 1 and can prove so by l'hopital's rule for single variable functions,sandwich theorem and by taylor's theorem!...am not here to disprove my tutor,i just want to know the l'hopital's rule for multivariate function,for personal pleasure,so i dont necessarily need completely proven facts and textbook quotes,I need a real discussion!I welcome eccentric thoughts and suggestions...i've been searching through google and got dropped here less than a week ago...Here's my thought,no proof,just speculation and am still looking into the facts to perfect my ideas,For a single variable function,l'hopital's rule differentiates a function f(x) with respect to x,so am thinking with multivariable functions I would take partial derivatives of first order followed by the mixed derivative...this in my case for now will be abstractly to show if a case is determinate or indeterminate,not that the result would be the real value itself,though it might be but not necessarily!You might not understand this so please dont take the trouble to post negative messages,am expecting an academically progressive discussion thank you!(adsbygoogle = window.adsbygoogle || []).push({});

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# L'Hôpítal's rule for multivariate functions!

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