Let's say I wish to find the limit of $\frac{x^{3}-y^{3}}{x^{2}+y^{2}}$ as $(x,y)\to (0,0)$. It is very simple to just let $y=mx;m\in\mathbb{R}$, and speculate that since the expression comes out to be $0$ on taking the limit and not a function of $m$, the limit may be $0$. Although, this obviously isn't very rigorous, because we're missing on infinitely many other paths of approach. So how would we prove this limit using Epsilon-delta. Any hints or ideas are appreciated. Thanks.
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5Hint. Note that $\frac{x^{3}-y^{3}}{x^{2}+y^{2}}=x\cdot\frac{x^{2}}{x^{2}+y^{2}}-y\cdot\frac{y^{2}}{x^{2}+y^{2}}$ – Robert Z Mar 23 '22 at 17:59
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2Does this answer your question? Use the $\epsilon - \delta$ definition to verify that $\lim_{(x,y)\to(0,0)}\frac{x^3-y^3}{x^2+y^2} = 0$? – 311411 Mar 23 '22 at 17:59
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1$|x|^3/(x^2+y^2)\le |x|.$ Another way for these type of problems is to use polar coordinates. The denominator becomes $r^2.$ – Ryszard Szwarc Mar 23 '22 at 17:59