Finding limits of 2 varible functions

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Homework Help Overview

The discussion revolves around finding the limit of a two-variable function, specifically F(x,y) = R, where R is defined as (x^2 + y^2) / (sqrt(x^2 + y^2 + 1) - 1. Participants are exploring the behavior of this function as it approaches the point (0,0).

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to analyze the limit by bounding R and questioning the applicability of one-variable limit methods, such as L'Hôpital's rule, to this two-variable context. Other participants suggest rationalizing the denominator as a potential approach to simplify the problem.

Discussion Status

Participants are actively discussing various methods to tackle the limit problem. Suggestions have been made to revisit the initial expression and rationalize the denominator, indicating a productive direction in the exploration of the problem.

Contextual Notes

There is an assumption that the limit is being investigated at the point (0,0), which is central to the discussion. The original poster expresses uncertainty about the applicability of techniques from single-variable calculus to this two-variable limit scenario.

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Homework Statement




Find the limit, if exist, otherwise show why it doesn't;

F(x,y) = R;

where R =

x^2 + y^2
------------------------
sqrt( x^2 + y^2 + 1) - 1


Attempt :

|R| = R

Since I know that |x| <= sqrt(x^2) <= sqrt(x^2 + y^2 + 1) = J

R <= J^2 + J^2 / (J + 1) = 2*J^2 / (J + 1)

Stuck here.

Also does the methods of 1 variable function to find limits, such as l'hopital rule apply to 2 variable? I assume not, since the book has not mentioned it.
 
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i assume it is at the limit at the point (0,0) you're trying to investigate?
 
Suggestion: Go back to the beginning and rationalize the denominator.
 
Billy Bob said:
Suggestion: Go back to the beginning and rationalize the denominator.

Perfect. Should have seen that. Thank you.
 

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