# Multivariable limit $\lim_{(x,y)\to(0,0)}\frac{\sqrt{4-x+y}-2}{7x-7y}$

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.

• Start multiplying numerator and denominator by $\sqrt{4-x+y}+2$. – N74 Oct 16 '16 at 21:24
• 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". – question Oct 16 '16 at 21:34

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