Proving that a limit of a two-variable function does not exist

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SUMMARY

The limit of the function (x^4 + y^4)/(x^3 + y^3) as (x,y) approaches (0,0) does not exist. Despite the server accepting 0 as the limit, the correct conclusion is that the limit diverges based on the approach taken. The discussion emphasizes the need to explore different paths, particularly along the line y=bx, to demonstrate that the limit does not converge to a single value. This challenge highlights the importance of understanding multivariable limits in calculus.

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  • Understanding of multivariable calculus concepts, specifically limits.
  • Familiarity with the epsilon-delta definition of limits.
  • Knowledge of polynomial functions and their behavior near critical points.
  • Ability to analyze functions along various paths in the Cartesian plane.
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  • Explore the epsilon-delta definition of limits in multivariable calculus.
  • Learn how to evaluate limits along different paths, particularly linear paths like y=bx.
  • Investigate examples of functions with non-existent limits in two variables.
  • Study the concept of continuity and its relationship to limits in multivariable functions.
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bellerevolte
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Find the limit as (x,y) -> (0,0) of (x^4 + y^4)/(x^3 + y^3)

This was a question from a recent homework set (class homework is done online), and the server accepted 0 as an answer. However, the actual answer is that the limit does not exist. My professor told us this afterwards and proposed that we find a way to prove that the limit indeed does not exist (I'm assuming this means to find a function from which the limit does not approach 0). But every function I have tried so far ends up making the limit 0.

Anyone up for a challenge? :)
 
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Consider the line y=bx. Can you find a value of b such that the function blows up at all nonzero values of x on this line?
 

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