Answer to Question #186262 in Calculus for noman

Question

Is the function h(u,v)=cos⁡(u)+e^nu+√(sinh⁡(u^2 )+√(v+e^nu ))   , Where n is second digit of your arid number for example if your arid number is 19-arid-12345 then choose n=2. Is is continuous? In case of yes then find the points of continuity. Explain briefly

Answer 1

$\displaystyle h(u,v)=cos⁡(u)+e^{2u}+\sqrt{\left(\sinh⁡(u^2 )+\sqrt{v+e^{2u}}\right)}\\ \textsf{For}\,\, h(u, v)\,\, \textsf{to continuous,}\,\, h(u, v)\\ \textsf{has to be defined}\\ \textsf{and for the function to be defined,}\\ v + e^{2u} > 0\,\, \\ \textsf{to make the the function}\\\textsf{in the square root positive}\\ \implies v > -e^{2u}\\ \textsf{Also, if}\,\, z = \sqrt{v + e^{2u}}\\ \sinh(u^2) + z > 0 \\ \sinh(u^2) > -z\\ u^2 > \arcsin\mathrm{h}\left(-\sqrt{v + e^{2u}}\right)\\ \textsf{But}\,\, v + e^{2u} > 0,\\ \implies u^2 > \arcsin\mathrm{h}\left(\sqrt{v + e^{2u}}\right) > \arcsin\mathrm{h}(0) = 0\\ \implies u > 0\\ \therefore h(u, v) \,\, \textsf{is continuous if and only if}\\ u >0\,\, \textsf{and}\,\, v > -e^{2u}.$