Question

Kim(x,y)->(0,0) sinx/y exist.is it true or false. Give reasons for your answer

Answer 1

Solution:

$\mathrm{Let's \ find\:two\:different\:paths\:to\:approach\:the\:point\:that}$

$\mathrm {\:gives\:different\:values\:for\:the\:limit}$

Case 1: $\lim_{(x,y)\rightarrow (0,0)}(\sin (\dfrac{x}{y}))$ along $x=0$ :

$=\sin (\dfrac{0}{y})=\sin (0)=0$

Case 2: $\lim_{(x,y)\rightarrow (0,0)}(\sin (\dfrac{x}{y}))$ along $x=y$ :

$=\sin (\dfrac{y}{y})=\sin (1)$

Thus, from case 1 and 2, we have that given limit diverges (or does not exist).

Hence, the given statement is false.