Chapter 11: Q23E (page 632)
Determine the set of points at which the function is continuous.
\(G(x,y) = \ln ({x^2} + {y^2} - 4)\)
\(\{ (x,y){\rm{ : }}{x^2} + {y^2} > 4\} \)
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Find the limit, if it exists, or show that the limit does not exist.
\(\mathop {lim}\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{xy}}{{\sqrt {{x^2} + {y^2}} }}\)
(a) Find the values of \(\frac{{\partial z}}{{\partial r}}\) and \(\frac{{\partial z}}{{\partial \theta }}\) if \(z = f(x,y)\), where \(x = r\cos \theta \) and \(y = r\sin \theta \).
(b) Show the equation\({\left( {\frac{{\partial z}}{{\partial x}}} \right)^2} + {\left( {\frac{{\partial z}}{{\partial y}}} \right)^2} = {\left( {\frac{{\partial z}}{{\partial r}}} \right)^2} + \frac{1}{{{r^2}}}{\left( {\frac{{\partial z}}{{\partial \theta }}} \right)^2}\).
Find and sketch the domain of the function.\(f(x,y) = \frac{{\sqrt {y - {x^2}} }}{{(1 - {x^2})}}\).
Graph and discuss the continuity of the function
\(f(x,y) = \left\{ \begin{aligned}{l}\frac{{sinxy}}{{xy}}, if xy \ne 0\\1, if xy = 0\end{aligned} \right.\)
Find the limit, if it exists, or show that the limit does not exist.
\(\mathop {lim}\limits_{\left( {x,y} \right) \to \left( {1,0} \right)} \frac{{xy - y}}{{{{(x - 1)}^2} + {y^2}}}\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
