Question

Give an example of a function of two variables whose limit at (0, 0) does not exist, that is

lim(x,y)→(0,0) f(x, y) does not exist. Explain also why the limit does not exist.

Answer 1

$lim_{(x,y)\to(0,0)}\frac{x²-y²}{x²+y²}$

This is an example of a function of two variables whose limit at (0, 0) does not exist.

The reason of non existence of the limit is discussed below.

Let us consider y = mx. So y$\to0$ as x $\to 0$ for any arbitrary m.

So the limit becomes

$lim_{x\to0}\frac{x²-m²x²}{x²+m²x²}$

= $lim_{x\to0}\frac{1-m²}{1+m²}$

= $\frac{1-m²}{1+m²}$

So the value of the limit varies as the value of m varies.

Hence the limit does not exist