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Using polar coordinates to evaluate a multivariable limit

  • Thread starter Jimmy25
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1. The problem statement, all variables and given/known data

When you substitute polar coordinates into a multivariable limit, do you treat theda as a constant when evaluating? (I know how to use polar coordinates to evaluate a limit but haven't learned what they are yet)

2. Relevant equations



3. The attempt at a solution
 

fzero

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1. The problem statement, all variables and given/known data

When you substitute polar coordinates into a multivariable limit, do you treat theda as a constant when evaluating? (I know how to use polar coordinates to evaluate a limit but haven't learned what they are yet)
Generally you wouldn't treat [tex]\theta[/tex] as a constant because the most important concept in taking a multivariable limit is that the limit doesn't exist if it depends on the path you take to the point.

As an example, consider

[tex]\lim_{(x,y)\rightarrow (0,0)} \frac{x^2y}{x^4+y^2} = \lim_{r->0} \frac{r\cos^2\theta\sin\theta}{r^2 \cos^4\theta + \sin^2\theta} = \lim_{r->0} \frac{r \cos^2\theta}{\sin\theta}.[/tex]

If we just take [tex]r\rightarrow 0[/tex], we find that this vanishes. However, if we also take [tex]\theta\rightarrow 0[/tex] at the same rate as r, we find [tex]\cos^2(0)=1[/tex]. Therefore the limit does not exist.
 

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