TITLE: $ \lim_{(x,y) \rightarrow (0,0)} (x^2+y^2)( \sin( \frac{1}{x^2+y^2} ))$
QUESTION [2 upvotes]: I was doing exercises on Howard Anton Calculus and I came across a problem which asks to find:

$$ \lim_{(x,y) \rightarrow (0,0)} (x^2+y^2)\sin\left( \frac{1}{x^2+y^2} \right).$$.

Intuitively we know the answer: $0$, but is there a step by step procedure that can be proposed as an argument ?

REPLY [0 votes]: Another possible approach is to notice that the function $f(x,y)=x^2+y^2$ tends to $0$ as $(x,y)$ goes to $(0,0)$. Then, you can reduce yourself to the limit 
$$\lim_{z \to 0} z \sin(\frac{1}{z})$$
with the change of variable $z=f(x,y)$, and we know that the above limit is $0$. The nice thing is that this approach works any time you have a "disguised" known limit of one variable.
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