<s>[INST] Continuity of $f(x,y) = \dfrac{\sin(x^2+y^2)}{x^2+y^2}$ at $(x,y) = (0,0)$ Continuity of $f(x,y) = \dfrac{\sin(x^2+y^2)}{x^2+y^2}$ at $(x,y) = (0,0)$
I need to specify the value of $f(x,y)$ which makes the given function continuous.
I had tried to give a arbitral relationship between $y$ and $x$, but as $(x, y)$ gets closer to $(0, 0)$ the arbitral movement cannot always be represented as a functional representation between $y$ and $x$.
How could I solve this problem rigorously?
 [/INST] For $\lim_{(x,y)\rightarrow (0,0)} \frac{sin(x^2 +y^2}{x^2+y^2}$, using $r^2=x^2+y^2$ we get $\lim_{r\rightarrow 0} \frac{sin(r^2)}{r^2}$=$\lim_{r \rightarrow 0} \frac{2 \cdot r \cdot \cos(r^2)}{2r}$=$\lim_{r \rightarrow 0} \cos(r^2)=1$, therefore $\lim_{(x,y)\rightarrow (0,0)} \frac{sin(x^2 +y^2}{x^2+y^2}=1$.
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