Esercizio 17 limiti in due variabili

Limiti in due variabili

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Esercizio 17  (\bigstar\bigstar\largewhitestar\largewhitestarr\largewhitestar\largewhitestar). Calcolare, se esiste, il seguente limite:

(1)   \begin{equation*} \lim_{(x,y)\to (0,0)} \;\dfrac{\sqrt{x}\sin y}{\left \vert x \right \vert + \left \vert y\right \vert }. \end{equation*}

 

Svolgimento. Osserviamo che per x\geq 0 e y\in \mathbb{R} vale quanto segue

    \[\left \vert \dfrac{\sqrt{x}\sin y}{\left \vert x\right \vert + \left \vert y\right \vert }\right \vert \leq \sqrt{x}\left \vert\dfrac{\sin y }{y}\right \vert\longrightarrow0,\quad \text{per $(x,y)\to (0,0)$}.\]

Si conclude che

    \[\boxcolorato{analisi}{ \lim_{(x,y)\to (0,0)} \;\dfrac{\sqrt{x}\sin y}{\left \vert x \right \vert + \left \vert y\right \vert }=0. }\]