Multi-variable limit using epsilon-delta technique

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SUMMARY

The discussion focuses on proving the limit of the multivariable function lim (3xy^2)/(x^2+y^2) as (x,y) approaches (0,0) using the epsilon-delta technique. The user initially struggles with the concept but finds clarity in converting the function to polar coordinates, which simplifies the limit evaluation. By expressing the function in terms of r (the radius in polar coordinates), it becomes evident that the limit approaches zero as r approaches zero. This method effectively demonstrates the limit's validity without ambiguity.

PREREQUISITES
  • Understanding of multivariable calculus concepts, particularly limits.
  • Familiarity with the epsilon-delta definition of limits.
  • Knowledge of polar coordinates and their application in calculus.
  • Basic algebraic manipulation skills for functions of multiple variables.
NEXT STEPS
  • Study the epsilon-delta definition of limits in detail.
  • Learn how to convert Cartesian coordinates to polar coordinates.
  • Practice solving limits of multivariable functions using polar coordinates.
  • Explore additional examples of limits in multivariable calculus for deeper understanding.
USEFUL FOR

Students of multivariable calculus, particularly those struggling with limit proofs, as well as educators seeking effective teaching methods for the epsilon-delta technique.

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Hi guys, I'm new to this forum, I am a mechanical engineering student, I am actually taking Multi-variable Calculus (Calculus III, on my college) this semester and I'm having a lot of trouble proving the limit of a multivariable equation using the epsilon-delta technique.

Example:

lim (3xy^2)/(x^2+y^2) x->(0,0)

I don't even understand what is about, I look the limit of the equation along x and y-axis y=mx and y=x^2 and found L=0 but how do I prove it is really 0.
The book show something about epsilo-delta technique but say's nothing about how using this technique, it look kind of random.

Thanks a lot Link-

PS. Sorry for my english, I am not a good english speaker or writter but at always I do my best :biggrin:
 
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The easiest way to do this problem is to rewrite the function in polar coordinates. Then it should be easy to see that as r->0, the function goes to zero.
 
StatusX said:
The easiest way to do this problem is to rewrite the function in polar coordinates. Then it should be easy to see that as r->0, the function goes to zero.

Thanks a lot.
 

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