Limit as 9x,y) approaches (0,0)

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In summary, the conversation is about justifying the existence of the limit of a function as (x,y) approaches (0,0) and finding the limit using the squeezing technique. The attempt at a solution includes finding the limit of f(x,0) and f(0,y), and then extending the evaluation to all linear trajectories towards the origin. Finally, the conversation discusses the possibility of the limit depending on the line of approach and the potential for the limit to not exist. A Wolfram Alpha link is provided for further reference.
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Jadehaan
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Homework Statement



Justify if the limit of the following function exists as (x,y) approaches (0,0). If it exists find the limit using the squeezing technique.
f(x,y)=(exy-1)/(x2+y2)

Homework Equations





The Attempt at a Solution



I found the limit of f(x,0) to approach 0
I found the limit of f(0,y) to approach 0
Since this is insufficient I found the limit of f(x,x)=(ex2-1)/2x2

Thanks for any help.
 
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  • #2
why not extend the evaluation to all linear trajectories towards the origin given by y = cx, for some real c

then evaluate the limit of
(lim x->0) g(x) = f(x,cx)
if it is indeterminate, try using L'hopitals rule

does the limit depend on c?
 
  • #3
Unfortunately, even if the limit, as you approach (0,0) along every line is the same, it might still be different for some curve- and so the limit might not exist.
 
  • #4
ok, but if you can show that dependent on the line of approach, the result differs, you have shown the limit does not exist
 

1. What is the limit as (x,y) approaches (0,0)?

The limit as (x,y) approaches (0,0) is the value that a function approaches as its inputs get closer and closer to (0,0). In other words, it is the value that the function is getting closer to but may never actually reach.

2. How do you calculate the limit as (x,y) approaches (0,0)?

To calculate the limit as (x,y) approaches (0,0), you can use the limit definition which involves evaluating the function at multiple points that get closer and closer to (0,0). Alternatively, you can use techniques such as substitution, algebraic manipulation, or L'Hôpital's rule.

3. What does it mean for a limit as (x,y) approaches (0,0) to not exist?

If the limit as (x,y) approaches (0,0) does not exist, it means that the function does not approach a single value as its inputs get closer to (0,0). This could be due to the function having a jump or a discontinuity at (0,0), or the limit may approach different values depending on the direction of approach.

4. Can a limit as (x,y) approaches (0,0) be finite?

Yes, a limit as (x,y) approaches (0,0) can be finite. This means that the function approaches a specific value as its inputs get closer to (0,0). However, the limit can also be infinite or not exist.

5. Why is the limit as (x,y) approaches (0,0) important in mathematics?

The limit as (x,y) approaches (0,0) is important in mathematics because it helps us understand the behavior of a function at a specific point. It can also be used to determine continuity, differentiability, and other properties of a function. Additionally, it is a fundamental concept in calculus and is used in various applications such as optimization, optimization, and physics.

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