Final answer:
The limit lim_(((x,y) \to (0,0))) (xy)/(\sqrt(x^2+y^2)) does not exist because approaching (0,0) from different paths (along the axes and the line y=x) yields different limit values.
Step-by-step explanation:
The question asks whether the limit lim_(((x,y) \to (0,0))) (xy)/(\sqrt(x^2+y^2)) exists as the variables (x, y) approach (0, 0). To determine the existence of this limit, you must check if the value of the limit is the same when approaching (0,0) from different paths. If the limit exists, it must be unique, regardless of the approach.
Let's investigate the limit by approaching (0,0) along the x-axis (where y=0), and along the y-axis (where x=0).
- Approach along x-axis (y=0): lim_(((x,0) \to (0,0))) (x*0)/(\sqrt(x^2+0^2)) = 0.
- Approach along y-axis (x=0): lim_(((0,y) \to (0,0))) (0*y)/(\sqrt(0^2+y^2)) = 0.
However, if we approach along y=x, we get a different scenario:
- Approach along y=x: lim_(((x,x) \to (0,0))) (x*x)/(\sqrt(x^2+x^2)) = lim_(((x,x) \to (0,0))) (x^2)/(x\sqrt(2)) = 1/\sqrt(2), which is not equal to 0.
Since we have different limits when approaching from different paths, the original limit does not exist. Therefore, the answer is No, the limit does not exist.