I'm trying to show that the limit as $(x,y)$ go to $(0,0)$ for the function $f(x,y) = sin( x + y )/( |x| + |y|)$ does not exist. I initially tried the path $y=2$ and $y=1$, but I don't think I can use these paths because they don't go through the origin. Any thoughts on which path to choose to show that this limit doesn't exist?
Try the paths
- $y=x$ with $x>0$; and
- $y=-x$ with $x>0$.
Both paths get arbitrarily close to the origin. Compute the limit restricted to each of those paths (you may need to apply L'Hôpital's rule). If you get two different results, the limit does not exist (because the limit can't depend on the path).