Math Problem Statement
Solution
The problem in the image is asking you to analyze the behavior of the function:
Specifically, it asks you to:
- Show that tends to 0 along any curve of the form for (where is any natural number except 3).
- Show that does not tend to 0 along the curve .
- Discuss what these results imply about the limit .
Step 1: Check the limit along for
Substitute into the expression for :
Now, analyze the behavior as . The leading term in the denominator will depend on :
- If , then dominates in the denominator as .
- If , then dominates.
In all these cases except , as , the numerator approaches 0 faster than the denominator. Hence, for any .
Step 2: Check the limit along
Now substitute into the function:
As , the term dominates the denominator (since vanishes faster than ). Thus:
Step 3: Conclusion on the limit
Since tends to different values (0 along some curves and does not approach 0 uniformly), we conclude that:
does not exist. The behavior of the function depends on the path taken toward the origin.
Would you like more details or have any further questions?
Here are some related questions for further practice:
- What happens to the limit of along the curve ?
- Can you verify the limit along for negative values of ?
- How would the function behave if the denominator had different powers of and ?
- What can you infer about the general behavior of rational functions like this one near singular points?
- How does this analysis extend to polar coordinates?
Tip: When evaluating multivariable limits, always try several paths to check if the limit is path-dependent or independent.
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Limits
Path Dependency
Formulas
f(x, y) = (x^3 * y^3) / (x^2 + y^4)
y = x^m
lim (x, y) → (0, 0) f(x, y)
Theorems
Limit existence along multiple paths
Path-dependent limits
Suitable Grade Level
University (Calculus III or Multivariable Calculus)
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