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

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:

$lim_{ (x,y)→(0,0)} ​ \dfrac{x²−y²} {x²+y²} ​$

The above function does not exist.

This is an example of a function of two variables whose limit at (0, 0)

The reason of non existence of the limit is;

If y = mx, and $y \to 0, x \to 0$ for any arbitrary m. The limit becomes

$lim_{x\to0}\dfrac{x²-m²x²}{x²+m²x²} \\ = lim_{x\to0}\dfrac{1-m²}{1+m²}\\ = \dfrac{1-m²}{1+m²}$

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

Hence the limit does not exist$.$