lim (x^3-1)/(x^3+2x^2y+xy^2-x^2-2xy-y^2)
(x,y)->(1,0)
alright i approached this at y=0. and got the limit as 3. where else can i approach this from to see if the limit really exist which im hoping it doesn't. i tried y=x-1 and that was a mess and i also tried y=mx which did nothing.
Hmmm. . . Looks like a removable singularity to me.
x^3 - 1 = (x-1)(x^2 + xy + y^2)
x^3+2x^2y+xy^2-x^2-2xy-y^2
= x(x+y)^2 - (x+y)^2
= (x-1)(x+y)^2
so, f(x,y) = (x^2+xy+y^2)/(x+y)^2 for x≠1
limit as (x,y) -> (1,0) = (1)/(1)^2 = 1
To determine if the limit exists at a specific point, you need to check if the function approaches the same value regardless of the direction from which you approach that point.
You have correctly approached the limit at (1,0) by considering y=0. To confirm if the limit exists, you can check other paths approaching the point.
Path 1: Approaching from y=x-1
Let's consider y=x-1. Substitute this into the given expression:
lim (x^3-1)/(x^3+2x^2(x-1)+x(x-1)^2-x^2-2x(x-1)-(x-1)^2) as x approaches 1.
Simplifying the expression yields:
lim (x^3-1)/(x^3+x^2-x^2-2x+2x-1-1) as x approaches 1.
lim (x^3-1)/(x^3-2) as x approaches 1.
Substituting x=1 into the expression:
(1^3-1)/(1^3-2) = 0/(-1) = 0.
The limit at (1,0) when approaching from y=x-1 does exist and is equal to 0.
Path 2: Approaching from y=mx
Now, let's consider the path y=mx, where m is any real number. Substitute this into the given expression:
lim (x^3-1)/(x^3+2x^2(mx)+x(mx)^2-x^2-2x(mx)-(mx)^2) as (x,y) approaches (1,0).
Simplifying the expression yields:
lim (x^3-1)/(x^3+2mx^3+x^3m^2-x^2-2mx^2-(mx)^2) as (x,y) approaches (1,0).
lim (x^3-1)/(x^3(1+2m+m^2)-x^2(1+2m+m^2)-(m^2)x^2) as x approaches 1.
Substituting x=1 into the expression:
(1^3-1)/(1^3(1+2m+m^2)-1^2(1+2m+m^2)-(m^2)1^2) = (0)/(1+2m+m^2-1-2m-2m^2-m^2) = 0/(-m^2) = 0.
The limit at (1,0) when approaching from y=mx exists for any value of m and is equal to 0.
Therefore, from these calculations, we can conclude that the limit at (1,0) exists and is equal to 0, regardless of the path taken to approach the point.