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Welp. I can't find anything helpful, so here we go.

I'm trying to find the multivariable limit (if it exists) of $$\lim_{(x,y)\to(0,0)}\frac{\sqrt{4-x+y}-2}{7x-7y}$$

Mathematica and Wolfram Alpha both say that it evaluates to $\frac{-1}{28}$. Unfortunately, I have no idea how to reach this conclusion - or any conclusion, really.

I've tried using polar substitution, approaching $(0,0)$ along the x- and y-axes, as $y = x$ and $y = -x$... Nothing works.

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    $\begingroup$ Start multiplying numerator and denominator by $\sqrt{4-x+y}+2$. $\endgroup$
    – N74
    Commented Oct 16, 2016 at 21:24
  • $\begingroup$ You should be able to make this work with polar substitution. Just expand the root with taylor... You should try it this way, as this is a much more general approach than multiplying with the "conjugate". $\endgroup$
    – question
    Commented Oct 16, 2016 at 21:34

1 Answer 1

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$$\lim _{ (x,y)\to (0,0) } \frac { \sqrt { 4-x+y } -2 }{ 7x-7y } =\lim _{ (x,y)\to (0,0) } \frac { y-x }{ 7\left( x-y \right) \left( \sqrt { 4-x+y } +2 \right) } =\\=\lim _{ (x,y)\to (0,0) } -\frac { 1 }{ 7\left( \sqrt { 4-x+y } +2 \right) } =-\frac { 1 }{ 28 } $$

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  • $\begingroup$ Oh my gosh, thank you. $\endgroup$
    – krourou2
    Commented Oct 16, 2016 at 22:40

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