Multivariable Calculus: When to use Squeeze Theorem to solve limits?

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SUMMARY

The discussion focuses on the application of the Squeeze Theorem in multivariable calculus, specifically in evaluating the limit lim (x,y) -> (0,0) [(xy^2)/(x^2+y^4)]. Participants conclude that while the limit does not exist according to the textbook, the Squeeze Theorem can be applied to show that the limit approaches 0 under certain conditions. The conversation highlights the importance of considering multiple paths when evaluating limits, as a single path may lead to misleading conclusions.

PREREQUISITES
  • Understanding of multivariable limits
  • Familiarity with the Squeeze Theorem
  • Basic knowledge of calculus concepts
  • Ability to analyze functions of two variables
NEXT STEPS
  • Study the Squeeze Theorem in detail
  • Explore examples of limits in multivariable calculus
  • Learn about path-dependent limits in multivariable functions
  • Investigate alternative methods for proving limits, such as epsilon-delta definitions
USEFUL FOR

Students and educators in calculus, particularly those focusing on multivariable calculus and limit evaluation techniques.

theBEAST
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Homework Statement


For example in:
lim (x,y) -> (0,0) [(xy^2)/(x^2+y^4)]

This limit does not exist (according to textbook), but if you use squeeze theorem since y^2<(x^2+y^4)
y^2/(x^2+y^4) <= 1 and therefore

0 <= (xy^2)/(x^2+y^4) <= x
as x--> 0
so
lim (x,y) -> (0,0) [(xy^2)/(x^2+y^4)] = 0?
 
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What happens if you look at y=sqrt(x)?
 
dirk_mec1 said:
What happens if you look at y=sqrt(x)?

I see what you mean but what if i was looking at a different limit and I got 0 every time while using different paths. I still couldn't prove it with squeeze theorem because what if there is just this one path that i never tried.
 

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