[tex]\displaystyle\lim_{(x,y)\to(0,0)}\frac{\left(x+23y)^2}{x^2+529y^2}[/tex]
Suppose we choose a path along the [tex]x[/tex]-axis, so that [tex]y=0[/tex]:
[tex]\displaystyle\lim_{x\to0}\frac{x^2}{x^2}=\lim_{x\to0}1=1[/tex]
On the other hand, let's consider an arbitrary line through the origin, [tex]y=kx[/tex]:
[tex]\displaystyle\lim_{x\to0}\frac{(x+23kx)^2}{x^2+529(kx)^2}=\lim_{x\to0}\frac{(23k+1)^2x^2}{(529k^2+1)x^2}=\lim_{x\to0}\frac{(23k+1)^2}{529k^2+1}=\dfrac{(23k+1)^2}{529k^2+1}[/tex]
The value of the limit then depends on [tex]k[/tex], which means the limit is not the same across all possible paths toward the origin, and so the limit does not exist.
