For each of the following evaluate the limit or show that the limit does not exist $\lim_{(x, y) \to (0,0)} \frac{\sin(x-y)}{||(x, y)||}$

Question

For each of the following evaluate the limit or show that the limit does not exist

$\lim_{(x, y) \to (0,0)} \frac{\sin(x-y)}{\|(x, y)\|}$

Solution:

$=\lim_{(x, y) \to (0,0)} \frac{\sin(x-y)}{\sqrt{x^2 + y^2}}$

If we let $y = x$, then

$=\lim_{x \to 0)} \frac{\sin(0)}{\sqrt{2x^2}} = 0$ [Wouldn't this be indeterminate $0/0$?]

If we let $y = -x$, then

$\lim_{x \to 0^+} \frac{\sin(2x)}{\sqrt{2x^2}} = \lim_{x \to 0^+} \frac{\sqrt{2}\sin(2x)}{2x} = \sqrt{2}$

and since they are different values we know the limit does not exist.

Could someone please explain

Answer 1

It wouldn't be indeterminate, remember that in a limit does not matter the value of the object in the point, only matters what happens as we approach it. So $\lim\limits_{x\rightarrow 0} 0/x^2=0$.

Answer 2

When $y=0$, then $$\lim_{x>0\rightarrow 0}\ \frac{\sin\ x}{x} =1 $$ and $$\lim_{x<0\rightarrow 0}\ \frac{\sin\ x}{-x} =-1 $$