Limits and Continuity
1. Evaluate the following limit.
solution
(a) \( \displaystyle{\lim_{(x,y)\to(1,-2)} \frac{xy}{x^2+y^2}} \)
(b) \( \displaystyle{\lim_{(x,y)\to(0,\pi/4)} x+\tan y - \sin
(x-y) } \)
Show that each of the following limits doesn't exist.
2.
\( \displaystyle{\lim_{(x,y)\to(0,0)} \frac{3x^2-y^2}{x^2+2y^2}} \)
solution
3.
\( \displaystyle{\lim_{(x,y)\to(0,0)} \frac{2xy}{x^2+y^2}} \)
solution
4.
\( \displaystyle{\lim_{(x,y)\to(0,0)} \frac{2xy^2}{x^2+y^4}} \)
solution
5. Evaluate the limit. \( \displaystyle{\lim_{(x,y)\to(0,0)}
\frac{5x^2y}{x^2+y^2}} \)
solution (not in
test 2)