Interesting Multivariable Limit Problem

[Liquid7800]

Homework Statement

lim (x,y)→(3,1) | (x²y - 6xy - x² + 6x - 9y - 9) / ( (x-3)⁴ + (y-1)² )

Homework Equations

none

The Attempt at a Solution

Since we have the indeterminate form '0/0' when we simply substitute (x, y) = (3, 1) we need to show the limit Does Not Exist.
we will show that the function approaches different values as (x, y) approaches (3, 1) along different paths in R²

My question is:
Would I need to factor/simplify this multivariable polynomial and binomial before iteratively setting x and y on some path to find a path such that f (x, y) approaches different values as (x, y) → (3, 1) along different paths, thus proving the limit does not exist.

The reason I am asking is the algebra is getting pretty ugly as I try to apply different paths...as I am I having trouble factoring the multivariable polynomial.

Any hints or ideas are appreciated...thank you very much as you all have been very helpful to me in the past

 

[Tedjn]

Wait, so is (x,y) approaching (0,0) or is (x,y) approaching (3,1)?

 

[Liquid7800]

sorry... it is supposed to be approaching (3,1)...I edited the post

 

[Tedjn]

Since we have the indeterminate form '0/0' when we simply substitute (x, y) = (3, 1) we need to show the limit Does Not Exist.

Be careful. My function may be indeterminate at one point, but that does not mean the function does not tend toward a single value from all directions. Why?

we will show that the function approaches different values as (x, y) approaches (3, 1) along different paths in R²

The limit may or may not exist (before further investigation). If you want to try showing that the limit does not exist, this is the correct method. Factoring is never necessary, although it may prove to be a useful way to identify good paths to try. This depends on the problem, though. In your case, what paths have you tried that led you into ugly algebra? Are their two simple paths you have not tried that would work instead? (I believe I've found two.)

 

[Liquid7800]

Be careful. My function may be indeterminate at one point, but that does not mean the function does not tend toward a single value from all directions. Why?

Sorry, what I meant to state was: if we have the form '0/0' then we must finding working paths to calculate the paths of x and y in the limit
thus, if we get two different working paths for each variable then the limit does not exist

...what paths have you tried that led you into ugly algebra

The ugly algebra was resulting from trying to simplify the entire multivariable equation and then applying vertical line paths and horizontal paths.
That is why I wondered if I needed to simplify the equation much like trying to simplify a single variable limit for solving algebraically.

 

[Tedjn]

If you are trying a vertical and horizontal path, wouldn't the equation simplify itself since one of the variables becomes a fixed constant?

 

[HallsofIvy]

Homework Statement

lim (x,y)→(3,1) | (x²y - 6xy - x² + 6x - 9y - 9) / ( (x-3)⁴ + (y-1)² )

Have you written this correctly? At x= 3, y= 1, the numerator is 9- 18- 9+ 18- 9- 9= -18, NOT 0. The denominator is 0 but the numerator is not.
Perhaps you meant
$$\frac{x^2- 5xy+ x^2+ 6x- 9y- 9}{(x- 3)^4+ (y- 1)^2}$$?

Homework Equations

none

The Attempt at a Solution

Since we have the indeterminate form '0/0' when we simply substitute (x, y) = (3, 1) we need to show the limit Does Not Exist.
we will show that the function approaches different values as (x, y) approaches (3, 1) along different paths in R²

My question is:
Would I need to factor/simplify this multivariable polynomial and binomial before iteratively setting x and y on some path to find a path such that f (x, y) approaches different values as (x, y) → (3, 1) along different paths, thus proving the limit does not exist.

The reason I am asking is the algebra is getting pretty ugly as I try to apply different paths...as I am I having trouble factoring the multivariable polynomial.

Any hints or ideas are appreciated...thank you very much as you all have been very helpful to me in the past

 

[Liquid7800]

All, I appreciate the help

I tried two paths
x=3 | y -> 1 and then another path y = x-2 | x-> 3 that produced two different limits showing that the limit DNE...which is what I wanted.

My main issue was recognizing when to simplify the equation and when not to...Simplifying the multivariable polynomial was quite difficult but luckily I didnt need to.

Again I appreciate the insight from you all...always the best