Question

Find the limit, if it exists. (if an answer does not exist, enter dne.) lim (x, y) → (0, 0) x4 − 20y2 x2 + 10y2

Answer 1

Traveling along the x-axis, we have

$\displaystyle\lim_{(x,y)\to(0,0)}\frac{x^4-20y^2}{x^2+10y^2}=\lim_{x\to0}\frac{x^4}{x^2}=\lim_{x\to0}x^2=0$

On the other hand, along the y-axis we get

$\displaystyle\lim_{(x,y)\to(0,0)}\frac{x^4-20y^2}{x^2+10y^2}=\lim_{y\to0}\frac{-20y^2}{10y^2}=\lim_{y\to0}(-2)=-2$

Therefore the limit doesn't exist.