Use the $\varepsilon - \delta$ definition of the limit to verify that $\lim_{(x,y)\to(2,5)} xy = 10$

Question

Use the $\varepsilon - \delta$ definition of the limit to verify that $\lim_{(2,5)\to(2,5)} xy = 10$. Hint: xy - 10 = (x-2)(y-5) + 5(x-2) + 2(y-5).

I’ve got as far as defining $\delta$ as $\frac{\varepsilon}{8}$ but can’t figure out how to formally prove it. Been stuck on it for days, could anyone shed some light? :)

Answer 1

Let $\delta = \min(1,\frac{\epsilon}8 )$, hence we know that $\delta^2 \le \delta$.

\begin{align} |xy-10| &\le |x-2||y-5| + 5|x-2| + 2|y-5|\\ &< \delta^2 + 5\delta + 2\delta \\ &\le 8\delta\\ &\le \epsilon \end{align}