In this problem we show that the function f(x, y) = 8x - y/x + y does not have a

Question

In this problem we show that the function f(x, y) = 8x - y/x + 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 = 2x. Find the limit In this case: lim_(x, 2x) rightarrow (0, 0) 8x - y/x + y = Now consider (x, y) rightarrow (0, 0) along the curve y = 3x. Find the limit in this case: lim_(x, 3x) rightarrow (0, 0) 8x - y/x + 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 = mx. for arbitrary m. Find the limit In this case: lim_(x, mx) rightarrow (0, 0) 8x - y/x + y = (Be sure that you can explain how this result also indicates that f has no limit as (x, y) rightarrow (0, 0).

Explanation / Answer

a) (8x-y)/(x+y) = (8x-2x)/(x+2x) = 6x/3x = 2

limit = 2

b)(8x-y)/(x+y) = (8x-3x)/(x+3x) = 5/4

limit = 5/4

c)limit = (8-m)/(1+m)

since the limit varies as m varies, limit doesnot exist

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