triclino Posted January 22, 2010 Share Posted January 22, 2010 Suppose that ; 1)[math] f: R^2\rightarrow R[/math] such that : [math]f(x,y) =xy[/math] 2)The Euclidian norm of a vector [math] v=(u_{1}, u_{2})[/math] is defined as : [math] ||v||_{Eu} =\sqrt{ u_{1}^2 +u_{2}^2}[/math] 3) The maxnorm of a vector [math] v=(u_{1},u_{2})[/math] is defined as : [math] ||v||_{max} = max( |u_{1}|,|u_{2}|)[/math] Where [math] u_{1},u_{2}[/math] belong to the real Nos R Then prove : [math] \lim_{(x,y)\to(1,1)} f(x,y) = 1[/math] ,with respect to both norms Link to comment Share on other sites More sharing options...
Amr Morsi Posted March 28, 2010 Share Posted March 28, 2010 What Euclidean Norm and Maxnorm have to do here? You can easily get the limit by substituting with x=1 and y=1, you will get lim f(x,y)=1*1=1. Link to comment Share on other sites More sharing options...
triclino Posted April 1, 2010 Author Share Posted April 1, 2010 What Euclidean Norm and Maxnorm have to do here? You can easily get the limit by substituting with x=1 and y=1, you will get lim f(x,y)=1*1=1. That is not a proof is there?? Link to comment Share on other sites More sharing options...
Amr Morsi Posted April 2, 2010 Share Posted April 2, 2010 O.K...... I was talking like that because f(x,y) is obviously continuous, and lim f(x,y), as (x,y) tends to (xo,yo), is f(xo,yo). If you need an exact proof, then find the left limit which will be f(xo - epsilon , yo - epsilon). And, find the right limit which will be f(xo + epsilon , yo + epsilon). Then, since both are equal then the limit is found. Link to comment Share on other sites More sharing options...
triclino Posted April 2, 2010 Author Share Posted April 2, 2010 O.K...... I was talking like that because f(x,y) is obviously continuous, and lim f(x,y), as (x,y) tends to (xo,yo), is f(xo,yo). If you need an exact proof, then find the left limit which will be f(xo - epsilon , yo - epsilon). And, find the right limit which will be f(xo + epsilon , yo + epsilon). Then, since both are equal then the limit is found. How can you prove that left and right limits are equal?? How do you know that f(x,y) is continuous at ([math]x_{o},y_{o}[/math]),without proving it Link to comment Share on other sites More sharing options...
Amr Morsi Posted April 2, 2010 Share Posted April 2, 2010 1. You will have to check the Left Limit for the variable x. It will be lim f(x,y)=xo*y 2. Check the Right Limit for the variable x. It will be also lim f(x,y)=xo*y. 3. Therefore, the Left Limit of f(x,y), as x tends to xo, is equal to the Right Limit. 4. Repeat the previous, as y tends yo. And, you will get the same results. 5. Then lim f(x,y), as (x,y) tends to (xo,yo), is equal to f(xo,yo). Note: There are some simple functions that can be considered continuous, unless it is required to prove so....... such as f(x)=x, f(x)=x^2. Link to comment Share on other sites More sharing options...
triclino Posted April 2, 2010 Author Share Posted April 2, 2010 1. You will have to check the Left Limit for the variable x. It will be lim f(x,y)=xo*y2. Check the Right Limit for the variable x. It will be also lim f(x,y)=xo*y. 3. Therefore, the Left Limit of f(x,y), as x tends to xo, is equal to the Right Limit. 4. Repeat the previous, as y tends yo. And, you will get the same results. 5. Then lim f(x,y), as (x,y) tends to (xo,yo), is equal to f(xo,yo). Note: There are some simple functions that can be considered continuous, unless it is required to prove so....... such as f(x)=x, f(x)=x^2. You mean that: IF [math]lim_{y\to y_{o}}(lim_{x\to x_{o}}f(x,y))=f(x_{o},y_{o})=x_{o}.y_{o}[/math]..................... THEN [math] lim_{(x.y)\to(x_{o},y_{o})}f(x,y) =f(x_{o},y_{o})=x_{o}.y_{o}[/math]??? Link to comment Share on other sites More sharing options...
Amr Morsi Posted April 2, 2010 Share Posted April 2, 2010 But, you shouldn't go the first step directly. You must use Left and Right limits (for both x and y). Link to comment Share on other sites More sharing options...
triclino Posted April 3, 2010 Author Share Posted April 3, 2010 But, you shouldn't go the first step directly. You must use Left and Right limits (for both x and y). Now if this is not another pack of mathematical nonsenses what is it then? You keep on writing non existing theorems and conclusions ,by simply imagining them. -1 Link to comment Share on other sites More sharing options...
mooeypoo Posted April 3, 2010 Share Posted April 3, 2010 Lose the attitude, triclino, we are a debate forum here and this is unacceptable. If you have a claim to make, make it in a civil matter, or don't make it at all. Link to comment Share on other sites More sharing options...
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