Exploring Limits: Approaching (0,0) from Different Paths

In summary, the conversation discusses a homework problem involving finding the limit of a function as (x,y) approaches (0,0). The participant suggests using a substitution method taught by their instructor, but their substitution is incorrect. The correct substitution is eventually determined to be y = mx2, which leads to a limit of 1/(1+m2). The conversation then discusses the implications of this result and concludes that the limit does not exist as it depends on the value of m. The conversation also mentions a typo in the original problem statement and expresses gratitude for the correction.
  • #1
reddawg
46
0

Homework Statement


By considering different paths, show that the given function has no limit
as (x,y) [itex]\rightarrow[/itex] (0,0).

f(x,y) = x4/(x4 + y4)

Homework Equations





The Attempt at a Solution


My instructor taught me this process a while back and am unsure if it fits for this problem:

let y=mx2

limit as (x, mx2) [itex]\rightarrow[/itex] (0,0) of
x4/(x4 + mx4)

I tried this method seeing as letting x=0 yields limit = 0 and letting y = 0 yields limit = 1 ?
 
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  • #2
reddawg said:

Homework Statement


By considering different paths, show that the given function has no limit
as (x,y) [itex]\rightarrow[/itex] (0,0).

f(x,y) = x4/(x4 + y4)

Homework Equations





The Attempt at a Solution


My instructor taught me this process a while back and am unsure if it fits for this problem:

let y=mx2

limit as (x, mx2) [itex]\rightarrow[/itex] (0,0) of
x4/(x4 + mx4)

I tried this method seeing as letting x=0 yields limit = 0 and letting y = 0 yields limit = 1 ?
Your substitution is incorrect. If y = mx2, then y4 ≠ mx4.
 
  • #3
Oh right, it would become m2x4

Therefore it's x4/(x4 + m2x4)

Factoring out x4 yields 1/(1+m2)

In similar problems this method proved that the limit depended on the value of m, therefore it did not exist. Is this the case here? I would think so.
 
  • #4
Yes, different values of that limit for different values of "m" mean that the limit depends upon the path. Recall that if a function has a limit then the value of f must be close to that limit. But this shows the value of f close to (0,0) on the line y=2x is different from the value on the line y= 3x.
 
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  • #5
reddawg said:
Oh right, it would become m2x4
No, that's wrong as well.
If y = mx2 (as you have in the OP), then y4 = (mx2)4, right?
reddawg said:
Therefore it's x4/(x4 + m2x4)

Factoring out x4 yields 1/(1+m2)

In similar problems this method proved that the limit depended on the value of m, therefore it did not exist. Is this the case here? I would think so.
 
  • #6
Mark44 said:
No, that's wrong as well.
If y = mx2 (as you have in the OP), then y4 = (mx2)4, right?


Actually, that was a typo on my part while identifying the problem statement. It is supposed to be
y2 not y4.

Thank you for taking the time to catch that mistake though.
 

FAQ: Exploring Limits: Approaching (0,0) from Different Paths

What is the concept of limits in mathematics?

Limits in mathematics refer to the behavior of a function as its input values approach a certain value. It is used to describe the behavior of a function at a given point or as it approaches infinity or negative infinity.

How is the limit of a function determined?

The limit of a function can be determined by evaluating the function at values close to the point in question and observing the trend of the output values. Alternatively, mathematical techniques such as L'Hopital's rule or algebraic manipulation can also be used to determine the limit of a function.

What does it mean for a limit to exist?

A limit is said to exist if the function approaches a specific value as the input values approach a certain point. This means that the function is defined and has a finite value at that point. If the function approaches different values from different directions, the limit is said to not exist.

Can a function have a limit at a point where it is not defined?

Yes, a function can have a limit at a point where it is not defined. This means that the function approaches a specific value as the input values approach the undefined point. However, it is important to note that the limit and the value of the function at that point are not necessarily the same.

What is the practical application of limits in real-world problems?

Limits have various applications in real-world problems, such as determining the maximum or minimum value of a function, calculating the rate of change, and modeling physical phenomena. They are also used in fields like economics, engineering, and physics to analyze and solve complex problems.

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