Question

2xy Use the two-path test in order to show that the limit lim does not exist. 0.0)yx2

Answer 1

lim_(x, y) rightarrow (0, 1) 2 (xy)/y^2 + x^2 Now for two - path we prove these is two path for this limit Let f(x) = lim_x, y rightarrow 0, 0 2(xy)/y^2 + x^2 Now put y = mx = lim_x rightarrow 0 2(x.mx)/m^2 x^2 + x^2 = lim_x rightarrow 0 2.m.x^2/x^2 (1 + m^2) = 2m/1 + m^2 Now put y = x^2 f(x) = lim_x rightarrow 0 2.(x^2 - x)/x^4 + x^2 = lim_x rightarrow 2 x^3/(x^4 + x^2) = lim_x rightarrow 2x/x^2 + 1) = 0 Their is two path present y = mx phi y = x^2 so, the limits not exit