∵lim(x->+∞,y->-∞)[(x-y)^2/e^(x-y)]
=lim(t->+∞)(t^2/e^t)(令t=x-y)
=lim(t->+∞)(2t/e^t)(∞/∞型極限,應用羅比達法則)
=lim(t->+∞)(2/e^t)(∞/∞型極限,應用羅比達法則)
=0
lim(x->+∞)(x/e^x)
=lim(x->+∞)(1/e^x)(∞/∞型極限,應用羅比達法則)
=0
lim(y->-∞)(ye^y)
=lim(y->-∞)[y/e^(-y)]
=lim(y->-∞)[-1/e^(-y)](∞/∞型極限,應用羅比達法則)
=0
∴l(xiāng)im(x->+∞,y->-∞)[(x^2+y^2)e^(y-x)]
=lim(x->+∞,y->-∞)[((x-y)^2-2xy)/e^(x-y)]
=lim(x->+∞,y->-∞)[(x-y)^2/e^(x-y)-2(x/e^x)(ye^y)]
=lim(x->+∞,y->-∞)[(x-y)^2/e^(x-y)]-2*lim(x->+∞,y->-∞)(x/e^x)*lim(x->+∞,y->-∞)(ye^y)
=0-2*0*0
=0.