Question

求极限lim{n[ln(n+1)-lnn] n→∞

求极限
lim{n[ln(n+1)-lnn] n→∞
lim(x^3+x^2)/(x-2)^2 x→2
limx^2sin1/x^2 x→0

Answer 1

① 等价无穷小量替换： ln(1+t) ~ t (t->0)
lim(n→∞) n[ln(n+1)-lnn]
=lim(n→∞) nln[(n+1)/n]
=lim(n→∞) nln(1+1/n)
=lim(n→∞) n*(1/n)
=1

②
∵ lim( x→2) (x^3+x^2) = 12 ; lim( x→2) 1/(x-2)^2 = +∞

lim( x→2) (x^3+x^2)/(x-2)^2 = +∞

③ 重要极限 【或等价无穷小量替换： sint ~ t (t->0)】

lim(x->0) x^2sin1/x^2

=lim(x->0) sin(1/x^2)/(1/x^2)

= 1