Question

有木有大神会这道题，在线等啊？

Answer 1

lim(x->0-) e^(1/x) =0 and |sin(3/x^2)|≤1
=>lim(x->0-) e^(1/x). sin(3/x^2) =0
lim(x->0-) [ e^(1/x). sin(3/x^2) + x.arctan(1/x) ]
=lim(x->0-) x.arctan(1/x)
=lim(x->0-) x . lim(x->0-)arctan(1/x)
=0.(-π/2)
=0

Answer 2

当x->0-时，e^(1/x)->0，|sin(3/x^2)|<=1，所以e^(1/x)*sin(3/x^2)->0
同理，|arctan(1/x)|<π/2，所以x*arctan(1/x)->0
根据极限的四则运算性质
lim(x->0-) [e^(1/x)*sin(3/x^2)+x*arctan(1/x)]=0