Can someone help me calculate the following limits?
1) $ \displaystyle\lim _ {x \to 0 , y \to 0 } \frac{\sin(xy)}{\sqrt{x^2+y^2}} $ (it should equal zero, but I can't figure out how to compute it ) .
2) $\displaystyle\lim_ {(x,y)\to (0,\frac{\pi}{2} )} (1-\cos(x+y) ) ^{\tan(x+y)} $ (it should equal $1/e$).
3) $ \displaystyle\lim_{(x,y) \to (0,0) } \frac{x^2 y }{x^2 + y^4 } $ (which should equal zero).
4) $\displaystyle \lim_{(x,y) \to (0,1) } (1+3x^2 y )^ \frac{1}{x^2 (1+y) } $ (which should equal $e^{3/2}$ ).
Any help would be great !
Thanks in advance !