I don't really know how to solve this, and I have seen a duplicate of this question else where which is this: Does $\lim_{(x,y)\to(0,0)}[x\sin (1/y)+y\sin (1/x)]$ exist?
There is only 1 answer and it doesn't really clear my doubt. Basically, what I know about squeeze theorem is this: If we have 3 functions let's say $f(x)$, $g(x)$ and $h(x)$, and the limit of each function is laid out like this:
$$\lim_{(x,y)\to(0,0)} f(x,y)\le\lim_{(x,y)\to(0,0)} g(x,y)\le\lim_{(x,y)\to(0,0)}h(x,y)$$
But in his answer, he said $$|f(x,y)|\le |x| +|y|$$
This doesn't make sense how can squeeze theorem apply here?
Best Answer
The squeeze theorem applies here because $0 \le |f(x,y)| \le |x| + |y|$ (i.e., the missing lower bound function is just $0$), so as $(x,y) \to (0,0)$, the lower bound is already $0$ and the upper bound goes to $0$.