How can I prove this limit does not exist?

[Addez123]

Homework Statement

$$lim \frac {xy -1} {y-1} as (x,y) -> (1,1)$$

Relevant Equations

None

If I set x = 1, I can cancel out y-1 and get limit = 1
Now if I approach from the x-axis the numerator will be smaller or bigger than the denominator, but how would you prove that that does not result in 1 when you reach (x,y) = (1,1)?

TL;DR: Textbook says limit does not exist, but I obviously found a limit and can't find any way of not reaching that limit.

 

[FactChecker]

You can now set x=y and get a different limit. To prove that limit, can you use L'Hospital's Rule?

 

[Mark44]

Homework Statement:: $$lim \frac {xy -1} {y-1} as (x,y) -> (1,1)$$
Relevant Equations:: None

If I set x = 1, I can cancel out y-1 and get limit = 1
Now if I approach from the x-axis the numerator will be smaller or bigger than the denominator, but how would you prove that that does not result in 1 when you reach (x,y) = (1,1)?

TL;DR: Textbook says limit does not exist, but I obviously found a limit and can't find any way of not reaching that limit.

These two-variable limits can be a lot trickier than single-variable limits. With the latter, all you need to do is to show that the limit exists whether you approach from the left or from the right. If the limit from either direction fails to exist, or if you get different one-sided limits, the two-sided limit doesn't exist.
With limits of functions of two variables, you need to show that the limit exists along any path, straight line, curved, whatever. If you get different values on different paths, or the limit doesn't exist along some path, the limit doesn't exist.

To prove that limit, can you use L'Hospital's Rule?

The limit you described can be evaluated without the use of L'Hopital.

 

[fresh_42]

Homework Statement:: $$lim \frac {xy -1} {y-1} as (x,y) -> (1,1)$$
Relevant Equations:: None

If I set x = 1, I can cancel out y-1 and get limit = 1
Now if I approach from the x-axis the numerator will be smaller or bigger than the denominator, but how would you prove that that does not result in 1 when you reach (x,y) = (1,1)?

TL;DR: Textbook says limit does not exist, but I obviously found a limit and can't find any way of not reaching that limit.

And what do you get for $x=\dfrac{n-1}{n}\, , \,y=\dfrac{n}{n-1}$?

 

[WWGD]

Edit: Or fix x=1 and approach along (1,y), then along (x,x). Much trickier to show limit actually exists.

 

[PeroK]

Or fix y=1 and approach along (x,1), then along (x,x). Much trickier to show limit actually exists.

The function is not defined for $y = 1$.