Determining the limit for function of x and y

In summary, the conversation discusses the limit of a function f(x,y) as (x,y) approaches (0,0). The general limit does not exist, but there are limits when approaching from certain directions. To prove the general limit does not exist, one can choose a convenient direction and show that for any ##\delta>0##, there are points on the line within distance ##\epsilon## from (0,0) where the values of f(x,y) differ by more than 1. The line y=0 is suggested as a potential direction.
  • #1
WK95
139
1

Homework Statement


For f(x,y) = (2x - y^2)/(2x^2 + y), what is the limit as (x,y)->(0,0)?

YomJXbB.png

Homework Equations

The Attempt at a Solution


From this image, it seems that the limit would be non-existent since on one side of the sheet, it goes up and up to infinity whereas from the other side, it plunges down to negative infinity.

How can I show that the limit is DNE analytically?
 
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  • #2
The general limit does not exist. However there are limits when one approaches (0,0) from certain directions, eg along the line y=x the limit is 2.

To prove the general limit does not exist, just pick a convenient direction, ie a line in the x-y plane, and then show that for any ##\delta>0##, two points can be found on the line, both within distance ##\epsilon## from (0,0), for which the values of f(x,y) differ by more than 1. The line y=0 looks promising.
 

FAQ: Determining the limit for function of x and y

1. What is a limit for a function of x and y?

A limit for a function of x and y is the value that a function approaches as the independent variables, x and y, get closer and closer to a specified point. It represents the behavior of a function near a given point.

2. How is the limit of a function of x and y calculated?

The limit of a function of x and y is typically calculated by evaluating the function at different points close to the specified point and determining if the values are approaching a specific value or if they are becoming more and more unpredictable. This is known as the limit definition of a function.

3. What does it mean when a limit does not exist for a function of x and y?

A limit does not exist for a function of x and y when the function approaches different values from different directions or when the function approaches infinity or negative infinity. This indicates that the behavior of the function near the specified point is not well-defined.

4. Can the limit of a function of x and y be defined at a discontinuity?

No, the limit of a function of x and y cannot be defined at a point of discontinuity. This is because at a point of discontinuity, the function has a sudden jump or break, and the values on either side of the point are not approaching the same value, making the limit undefined.

5. How is the limit of a function of x and y used in calculus?

The limit of a function of x and y is an important concept in calculus as it is used to determine the derivative and integral of a function. It helps us understand the behavior of a function near a given point and can also be used to determine the continuity of a function at a point.

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