Show that the limit does not exist for the function
Lim (x,y)➡(1,-2): (4x^3y^2 - 2xy^5+ 7y-1)/ ( 3y-y^4 + 3x^3)
To show that the limit does not exist for the given function, we need to find two different paths that approach the point (1, -2) such that the limit of the function along these paths is not the same.
Let's consider two different paths:
1. Path 1: Approach along the x-axis (y = 0)
Taking the limit as x approaches 1 while y remains 0, we have:
lim (x->1, y->0) (4x^3y^2 - 2xy^5 + 7y - 1) / (3y - y^4 + 3x^3)
Substitute y = 0 into the function:
lim (x->1) (-1) / (3(0) + 3x^3)
= -1 / (3x^3)
As x approaches 1, the value of -1 / (3x^3) approaches -1/3.
2. Path 2: Approach along the y-axis (x = 0)
Taking the limit as y approaches -2 while x remains 0, we have:
lim (x->0, y->-2) (4x^3y^2 - 2xy^5 + 7y - 1) / (3y - y^4 + 3x^3)
Substitute x = 0 into the function:
lim (y->-2) (7y - 1) / (3y - y^4)
As y approaches -2, the value of (7y - 1) / (3y - y^4) approaches 3.
Since the limit along the x-axis (path 1) is -1/3 and the limit along the y-axis (path 2) is 3, these two limits are different. Therefore, the limit does not exist for the given function as (x, y) approaches (1, -2).