
limits and continuity
1a)Do these functions have limits.If the limit exists, find it with justification, if not explain why not
i) f(x,y) =x²y²/x²+y²
ii) f(x,y) =x³y³/x²+y²
iii) f(x,y) =xy/x+y
iv) f(x,y) =1√(1x²)/x²+ xy+y²
v) f(x,y) =y³x/y^6+x²
im having problems finding the limits especially iii) and iv). i know that you can prove a limit doesn't exist by demonstration but how do you justify it if it does exist
b) Let f be a real valued function of two real variables defined by:
f(x,y) ={2x sin y/ x²+y² if y >x
sin(x+y) if y≤x
Determine the points at which f is continuous.

Let's look at (iii). I claim the limit is zero: that is, for all epsilon > 0 there exists delta > 0 such that if 0 < sqrt(x^2+y^2) < epsilon then  xy(x+y)  < delta.
I claim we can take delta = epsilon. If 0 < sqrt(x^2+y^2)<epsilon, then x and y are less than epsilon and one of them, say x, is nonzero. Then x.y/(x+y < epsilon. x/(x + epsilon) < epsilon as claimed.