(a) eᵏ=1+k+k²/2!+...
e^(-x²-y²)=1-(x²+y²)+(x²+y²)²/2!-...
Therefore e^(-x²-y²)-1≈-(x²+y²).
So ( e^(-x²-y²)-1)/(x²+y²)→-1.
arctan(-1)=-π/4.
(b) cos(2xy)-1=2cos²(xy)-1-1=
2(cos²(xy)-1)= 2(cos(xy)+1)(cos(xy)-1)
Therefore:
(cos(2xy)-1)/(1-cos(xy))=-2(cos(xy)+1)→-4 in the limit as xy→2π, cos(2π)=1.
(c) 3xy/(x²+9y²).
Let x=y=h: expression becomes 3h²/10h²→3/10 as h→0.