Thread: continuity

show that the following function is continuos at (0,0)

$\displaystyle
f(x,y) = \left\{ \begin{array}{l}
\frac{{xsen(x)}}{{\sqrt {x^2 + y^2 } }}\,\,\,if(x,y) \ne (0,0) \\
0\,\,\,if\,(x,y) = (0,0) \\
\end{array} \right.

$

i need to know how to do this kind of problems, with this example i think i may be able to do other problems of the same kind

thanks

To aswering the question it is necessary to valuate the...

$\displaystyle \lim_{(x,y) \rightarrow (0,0)} \frac{x\cdot \sin x}{\sqrt{x^{2} + y^{2}}}$ (1)

At this scope we use polar coordinates...

$\displaystyle x= \rho\cdot \cos \theta$

$\displaystyle y= \rho\cdot \sin \theta$ (2)

... and limit (1) becomes...

$\displaystyle \lim_{\rho \rightarrow 0} \cos \theta\cdot \sin (\rho\cdot \cos \theta)$ (3)

Bur the limit (3) is 0 independently from $\displaystyle \theta$ so that the function You have proposed is continous in $\displaystyle (0,0)$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$