I. Determine if the function f(x,y) given below has a limit along all linear and

Question

I. Determine if the function f(x,y) given below has a limit along all linear and parabolic paths near the origin. What can you conclude about the continuity of the function at (0, 0)?

f(x,y)= { 0, if x=0 and y=0

Graph the function on a computer and turn in a printout which shows the surface at (0, 0).
Does the graph agree with your conclusion about continuity above?

Explanation / Answer

Choose the linear path x=y: f(x,y)=y^3/(2y^4+3y^2) = y^3/(y^2)*(2y^2+3) = y/(2y^2+3) =>0 Along a linear path, the functional value is equal the limit, implying continuity. But, Choose the parabolic path x^2=y: f(x,y)=y^2/(2y^2 +3y^2)=y^2/5y^2 => 1/5 Hence, along some (parabolic) path, the limit exists. However, the function is not continuous since the value at the point and the limit at that point are not equal (i.e. 0 does not equal 1/5). Thus the function is not continuous.

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