Question

Mostre, usando a definição de limite, que lim
(x,y)→(1,2) (x + 2y) = 5.

Answer 1

Veamos

$\lim\limits_{(x,y)\to (1,2)}x+2y=5\Longleftrightarrow\\ \\ \\ \forall\epsilon\ \textgreater \ 0\,,\,\exists\delta\ \textgreater \ 0:\|(x,y)-(1,2)\|\ \textless \ \delta\Rightarrow|x+2y-5|\ \textless \ \epsilon\\ \\ \\ \\ \text{de }\|(x,y)-(1,2)\|\ \textless \ \delta\text{ se deduce: }\\ \\ |x-1|\ \textless \ \delta \;,\; |y-2|\ \textless \ \delta\\ \\ \text{adem\'as }|x+2y-5|=|(x-1)+2(y-2)|\leq |x-1|+2|y-2|\\ \\ |x+2y-5|\leq 3\delta\\ \\ \\ \text{Entonces se puede tomar }\delta =\dfrac{\epsilon}{3}\text{ con lo que\,termina la prueba}\\ \\ \\ \hspace*{9cm}\square\square\square$