Question

Find the limit, if it exists, or show that the limit does not exist.

\(\mathop {lim}\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }}\)

Short answer

The limit of the given function exists.

The value of the limit of the function is: \(\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }} = 0\).

Step-by-step solution

Check whether the limit exist or not

Consider the function \(\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }}\)

Convert the function into polar coordinates

The following conversations can be used for polar coordinates.

\(\begin{aligned}{l}{x^2} + {y^2} = {r^2}\\rcos\theta = x\\rsin\theta = y\end{aligned}\)

\(\begin{aligned}{l}\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }} = \mathop {\lim }\limits_{r \to 0} \frac{{(r\cos \theta )(r\sin \theta )}}{{\sqrt {{r^2}} }}\\\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }} = \mathop {\lim }\limits_{r \to 0} \frac{{({r^2}\cos \theta )(\sin \theta )}}{r}\\\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }} = \mathop {\lim }\limits_{r \to 0} (r\cos \theta \sin \theta )\end{aligned}\)

The function\(f(x,y) = \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }}\) is polar function. So it is continuous at everywhere.

So the limit is exists.

Find the limit

\(\begin{aligned}{l}\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }} = \mathop {\lim }\limits_{r \to 0} (r\cos \theta \sin \theta )\\\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }} = (0)(\cos \theta \sin \theta )\\\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }} = 0\end{aligned}\)

Therefore, the given function is continuous. The limit is exists. The value of the limit of the function is: \(\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }} = 0\).