Question

Use polar coordinates and the squeeze theorem to find lim(x,y)rightarrow(0,0)x3-y3/x2+y2 Suppose f(x) = {x3 - y3/x2 + y2 if (x, y) (0,0) 0 if (x,y) = (0,0) Is f(x,y) continuous at (0,0)? Justify your answer In polar coordinates, f(x, y) = r3(cos3(theta) - sin3(theta))/r2 = r(cos3(theta) - sin3(theta)). So Since is contineous at (0,0) . It follows by the squeeze theorem that lim(x,y)rightarrow(0,0)x3 - y3/x2 + y2 = 0. Since f(0,0) = 0 = lim(x,y)rightarrow(0,0)f(x,y), f is continuous at (0,0).

Show transcribed image text Use polar coordinates and the squeeze theorem to find lim(x,y)rightarrow(0,0)x3-y3/x2+y2 Suppose f(x) = {x3 - y3/x2 + y2 if (x, y) (0,0) 0 if (x,y) = (0,0) Is f(x,y) continuous at (0,0)? Justify your answer In polar coordinates, f(x, y) = r3(cos3(theta) - sin3(theta))/r2 = r(cos3(theta) - sin3(theta)). So Since is contineous at (0,0) . It follows by the squeeze theorem that lim(x,y)rightarrow(0,0)x3 - y3/x2 + y2 = 0. Since f(0,0) = 0 = lim(x,y)rightarrow(0,0)f(x,y), f is continuous at (0,0).