Show, with the definition, that $\lim_\limits{ (x,y) \to (0,0)} x\sin\frac{1}{y} + y\sin\frac{1}{x}$ exist

Question

Show, with the definition, that $\lim_\limits{ (x,y) \to (0,0)} x\sin\frac{1}{y} + y\sin\frac{1}{x} $ exist; $(x,y) \in \mathbb{R^2}-\{(0,0)\}$.

I think the limit is zero because for $||(x,y)|| < \delta$, $$||x\sin\frac{1}{y} + y\sin\frac{1}{x}|| \leq ||(x,y)|| \cdot||(\sin\frac{1}{y},\sin\frac{1}{x})|| \leq \sqrt{2} ||(x,y)||< \sqrt{2}\delta.$$ It is sufficient to define that $\sqrt{2}\delta = \epsilon$

I am not certain of what I did so far. Is there anyone who can give me a hint to solve the problem?

Answer 1

You guessed correctly and that was crucial to make the next step. Let $\varepsilon>0$ and we shall find $\delta>0$ such that for all $(x,y)$ on the domain of $f$ such that $\|(x,y)-(0,0)\|_2<\delta$ we get $|f(x,y)-0|<\varepsilon$ (here $\|\cdot\|_2$ denotes the Euclidean metric). Indeed, if $\delta=\varepsilon/2>0$ then for all $(x,y)$ on the domain of $f$ such that $\|(x,y)-(0,0)\|_2<\delta$ we get: $$|f(x,y)-0|\leq |x||\sin(1/y)|+|y||\sin(1/x)|\leq |x|+|y|\leq \|(x,y\|_2+\|(x,y)\|_2<2\delta=\varepsilon.$$