Question

Find Continuity of
f(x,y)=y/√(x^2+y^2). at (x,y)=(0,0)

Answer 1

$\lim\nolimits_{{(x,y)\to(a,b)}} f(x,y)= f(a,b)$  is the definition of continuity. Now take the limit from the x-axis

$\lim\nolimits_{{(x,y)\to(0,0)}} f(x,0)=\lim\nolimits_{{(x,y)\to(0,0)}} \frac{0}{\sqrt{(x^2+0)}}=0\\$

Now take the limit from x=y

$\lim\nolimits_{{(x,y)\to(0,0)}} f(x,x)=\lim\nolimits_{{(x,y)\to(0,0)}} \frac{x}{\sqrt{(x^2+x^2)}}\\ \lim\nolimits_{{(x,y)\to(0,0)}} \frac{x}{\sqrt{(2x^2)}}= \frac{1}{\sqrt{2}}\\$

Limit from the x-axis is 0 but the limit from x=y line is $\frac{1}{\sqrt{2}}$ therefore the limit does not exist which means at (0,0) the funtion f(x,y) is discontinuous since limit f(x,y) does not equal to f(0,0)