Question

Find the limit, if it exists, or type dne if it does not exist.

Answer 1

$\displaystyle\lim_{(x,y)\to(0,0)}\frac{\left(x+23y)^2}{x^2+529y^2}$

Suppose we choose a path along the $x$-axis, so that $y=0$:

$\displaystyle\lim_{x\to0}\frac{x^2}{x^2}=\lim_{x\to0}1=1$

On the other hand, let's consider an arbitrary line through the origin, $y=kx$:

$\displaystyle\lim_{x\to0}\frac{(x+23kx)^2}{x^2+529(kx)^2}=\lim_{x\to0}\frac{(23k+1)^2x^2}{(529k^2+1)x^2}=\lim_{x\to0}\frac{(23k+1)^2}{529k^2+1}=\dfrac{(23k+1)^2}{529k^2+1}$

The value of the limit then depends on $k$, which means the limit is not the same across all possible paths toward the origin, and so the limit does not exist.