Answer(s) and explanation please In this problem we show that the function f(x,

Question

Answer(s) and explanation please

In this problem we show that the function f(x, y) = 7x^2 - y/x^2 + 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 = 3x^2. Find the limit in this case: lim_(x, 3x^2) rightarrow (0, 0) 7x^2 - y/x^2 + y = 1 Now consider (x, y) rightarrow (0, 0) along the curve y = 4x^2. Find the limit in this case: lim(x, 4x^2) rightarrow (0, 0) 7x^2 - y/x^2 + 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^2, for arbitrary m. Find the limit in this case: lim_(x, mx^2) rightarrow (0, 0) 7x^2 - y/x^2 + 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

f(x,y) = 7x2 - y/x2 + y

lim(y tends to o){lim f(x,y) x tends to 0}

1. o

2 7x2 -4x2 /x2 + 4x2 = 3x2 /5x2 for x = 0 then 0

3. 7x2 - mx2 /x2 + mx2 = at x = 0 mx2 = y = 0

on substituting first wrt y value and then with x and applying those values we get answers as 0

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