当x->0,y->0,求(1-cos(x^2+y^2))/((x^2+y^2)e(x^2y^2))的极限,我知道极限是0,但是不知道怎么算出来的了!

Question

当x->0,y->0,求(1-cos(x^2+y^2))/((x^2+y^2)e(x^2y^2))的极限,我知道极限是0,但是不知道怎么算出来的了!

Answer 1

lim(1-cos(x^2+y^2))/((x^2+y^2)e(x^2y^2))=
lim(1-cos(x^2+y^2))/(x^2+y^2)*lim1/e(x^2y^2)
=lim(1-cos(x^2+y^2))/(x^2+y^2)
=lim(1-cosa)/a=(用L'Hospital Rule)
=limsina/a
=1
就是把1/e(x^2y^2)先分离
x->0,y->0,a=x^2+y^2->0.