How can I prove this limit does not exist?

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In summary: So, we cannot approach along that path.In summary, the conversation discusses the difficulty in determining the limit of a function of two variables, with the example of $$lim \frac {xy -1} {y-1} as (x,y) -> (1,1)$$. The textbook claims that the limit does not exist, but the speaker has found a limit of 1 by setting x = 1 and canceling out y-1. They then question how to prove that this limit does not result in 1 when approaching from different paths. The possibility of using L'Hopital's Rule is also mentioned, but it is stated that the limit can be evaluated without it.
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Addez123
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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.
 
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  • #2
You can now set x=y and get a different limit. To prove that limit, can you use L'Hospital's Rule?
 
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Addez123 said:
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.
FactChecker said:
To prove that limit, can you use L'Hospital's Rule?
The limit you described can be evaluated without the use of L'Hopital.
 
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Addez123 said:
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}##?
 
  • #5
Edit: Or fix x=1 and approach along (1,y), then along (x,x). Much trickier to show limit actually exists.
 
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  • #6
WWGD said:
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##.
 

1. How do I know if a limit does not exist?

To prove that a limit does not exist, you need to show that the function has different values when approaching the limit from different directions. This can be done by evaluating the limit from the left and right sides and showing that they are not equal.

2. Can I use a graph to prove a limit does not exist?

Yes, a graph can be a helpful tool in proving that a limit does not exist. If the graph has a "jump" or a "hole" at the point of the limit, then the limit does not exist. Additionally, if the graph approaches different values from different directions, this also proves that the limit does not exist.

3. What is the difference between a one-sided and two-sided limit?

A one-sided limit only considers the values of the function as it approaches the limit from one side (either the left or right). A two-sided limit looks at the values of the function as it approaches the limit from both the left and right sides. In order for a two-sided limit to exist, the one-sided limits must be equal.

4. Can I use algebra to prove a limit does not exist?

Yes, algebra can be used to prove that a limit does not exist. If you can manipulate the function algebraically to show that the limit approaches different values from different directions, then this proves that the limit does not exist.

5. What is the importance of proving that a limit does not exist?

Proving that a limit does not exist is important because it helps us understand the behavior of a function at a specific point. It also allows us to identify any discontinuities or holes in the graph of the function. This information is crucial in many mathematical and scientific applications.

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