In this problem we show that the function fx y 5x2 y x2

In this problem we show that the function f(x, y) = 5x2 - y / x2 + y does not have a limit as (x, y) rightarrow (0,0). Suppose that we consider (x, y) rightarrow (0, 0) along the curve y = 2x2. Find the limit in this case: lim (x, 2x2) rightarrow (0, 0) 5x2 - y/x2 + y = Now consider (x, y) rightarrow (0, 0) along the curve y = 3x2. Find the limit in this case: lim (x, 2x2) rightarrow (0, 0) 5x2 - y/x2 + y = Note that the results from (a) and (b) indicate that f has no limit as (x, y) rightarrow (0,0) (be sure you can explain why!). To show this more generally, consider (x, y) rightarrow (0,0) along the curve y = mx2, for arbitrary m. Find the limit in this case: lim (x, mx2) rightarrow (0, 0) 5x2 - y/x2 + y = (Be sure that you can explain how this result also indicates that f has no limit as (x, y) rightarrow (0,0).

Solution

a part. Just substitute 2x^2 in for y. This gives you 3/3 so the limit is 1. b part. Substitute 3x^2 in for y. This gives you 2/4 so the limit is 1/2. c part. Since the limits along two different curves is different the limit as (x,y) approaches (0,0) can not exist. The m x^2 bit follows similarly. An easier way would be to approach (0,0) along x axis, y = 0, and then along y axis, x = 0.