1.lim x->无穷 x*sin(2x/(x^2+1)) 2

题目

1.lim x->无穷 x*sin(2x/(x^2+1)) 2
2.lim x->正无穷 ((x^2-1)/(x^2+1))^(x^2) e^-2

答案

1.
lim x->无穷 x*sin(2x/(x^2+1))
令x=1/t, t->0
得
lim t->0 sin((2/t)/((1/t)^2+1))/t
=lim t->0 sin((2t)/((1+t^2))/t
=lim t->0 ((2t)/((1+t^2))/t
=2
2. lim x->正无穷 ((x^2-1)/(x^2+1))^(x^2)
= lim x->正无穷 ((1+(-2)/(x^2+1))^(x^2)
=e^(limx->正无穷(-2)(x^2)/(x^2+1) )
=e^(-2)