Question

求所有题极限

Answer 1

Answer 2

(1)
x->0

tanx~ x+(1/3)x^3
sinx ~ x-(1/6)x^3

tanx - sinx ~ (1/2)x^3
ln(1-x^3) ~ -x^3
lim(x->0) (tanx-sinx)/ln(1-x^3)
=lim(x->0) (1/2)x^3/(-x^3)
=-1/2
(2)
x->0

e^x -1 ~ x
(e^x -1).sinx ~ x^2
1-cosx ~ (1/2)x^2
lim(x->0) (e^x-1)sinx/(1-cosx)
=lim(x->0) x^2/((1/2)x^2)
=2
(3)
lim(h->0) [ln(x+h) -lnx] /h (0/0)

=lim(h->0) 1/(x+h)
=1/x
(4)
x->0

(1-2x^3)^(1/3) ~ 1- (2/3)x^3
(1-2x^3)^(1/3) -1 ~ -(2/3)x^3
arctan(x^3) ~ x^3
lim(x->0) [(1-2x^3)^(1/3) -1)/ arctan(x^3)
=lim(x->0) -(2/3)x^3/ x^3
=-2/3
(5)
lim(x->a) (e^x-e^a)/(x-a) (0/0)
=lim(x->a) e^x
=e^a

Answer 3

以上