2 calculus qestion

qestion 1
If fsubx(xsub0, ysub0) and fy(xsub0, ysub0) both exist, then f is continuous at (xsub0, ysub0). Prove or disprove

Hint consider the function defined by f(x,y) = (xy)/(x^2+y^2) , if (x,y) is not equal (0,0) or f(x,y) =0.

question 2
Let f(x,y) = (sin^2 (x-y))/ (abs(x) +abs(y) ).
Prove that lim as (x,y) approach (0,0) of f(x,y)=0.

Hint: for all real numbers m and n. abs(sin(m+n)) smaller than or equal to abo(m+n) smaller than or equal to abs(m) +abs(n).

note: abs(m) means absolution value of m.

Thank you very much

Quote:

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Originally Posted by littlemu

qestion 1
If fsubx(xsub0, ysub0) and fy(xsub0, ysub0) both exist, then f is continuous at (xsub0, ysub0). Prove or disprove

Hint consider the function defined by f(x,y) = (xy)/(x^2+y^2) , if (x,y) is not equal (0,0) or f(x,y) =0.

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False. Use the example the book gives.

The function f(x,y) is defined on an open disk containing (0,0). Which means f is continous if and only if,
lim [(x,y)-->(0,0)] f(x,y) = f(0,0)=0

But if you choose the path x=y both approaching zero the limit of f(x,y) is then 1! It must always be zero if it continous.

This is Mine 55:):)th Post!!!

I shall use my own inequality which is simple but useful.

Let x,y>=0 then sqrt(x+y) <= sqrt(x)+sqrt(y)