Saturday, May 25, 2019

IGNOU MTE07 2019 SOLVED ASSIGNMENT 3(a)

Put

\begin{array}{l} x = r\cos \theta \\ y = r\sin \theta \end{array}

we obtain
\left| {f\left( {x,y} \right) - f\left( {0,0} \right)} \right| = \left| {\frac{{2{r^3}{{\cos }^3}\theta + 5{r^3}{{\sin }^3}\theta }}{{{r^2}}}} \right|

=\left| {\frac{{3{r^3}{{\sin }^3}\theta }}{{{r^2}}}} \right|
\begin{array}{l} \le 3r{\left| {\sin \theta } \right|^3}\\ \le 3r\\ \le 3r\sqrt {{x^2} + {y^2}} \end{array}

Let  \in > 0 be given and \delta = (1/3) \in .

Then \left| {f\left( {x,y} \right) - f\left( {0,0} \right)} \right|< \in , when

when \sqrt {{x^2} + {y^2}} < \delta

or

\left| {f\left( {x,y} \right) - f\left( {0,0} \right)} \right|< \in

when

\sqrt {{{(x - 0)}^2} + {{(y - 0)}^2}} < \delta

Hence the given function is continuous at (0,0).



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