Question

Is the function h(u,v)=cos⁡(u)+e^nu+√(sinh⁡(u^2 )+√(v+e^nu )), Where n=5 Is is continuous? In case of yes then find the points of continuity. Explain briefly

Answer 1

Checking continuity at (0,0)

$\lim _{x\to \:0}\left(\cos \left(x\right)+e^5x+\sqrt{\sinh \left(x^2\right)}+\sqrt{x+e^5x}\right)$

$=\cos \left(0\right)+e^5\cdot \:0+\sqrt{\sinh \left(0^2\right)}+\sqrt{0+e^5\cdot \:0}$

$\lim\limits_{u,v \to (0,0)}h(u,v)=\lim\limits_{u,v \to (0,0)} cos⁡(u)+e^nu+ \sqrt{(sinh⁡(u^2 )} + \sqrt{(v+e^nu ))}= 1$

h is contiuous at (0,0)

Since cos(u) is continuous on R h(u,v) is continuous on R