Question

求下列极限

求下列极限求下列极限，

Answer 1

(1)
x->0
arcsinx ~ x +(1/3)x^3
x-arcsinx ~ -(1/3)x^3
lim(x->0) (x-arcsinx)/(sinx)^2
=lim(x->0) -(1/3)x^3/x^2
=0
(2)
lim(x->0) ln(1+x^2)/(secx-cosx)
=lim(x->0) cosx. ln(1+x^2)/(1-(cosx)^2)
=lim(x->0) cosx. ln(1+x^2)/ (sinx)^2
=lim(x->0) ln(1+x^2)/ (sinx)^2
=lim(x->0) x^2/ x^2
=1
(3)

lim(x->0+) lntan7x/lntan2x (0/0)
=lim(x->0+) [7(sec7x)^2/tan7x]/[2(sec7x)^2/tan2x]
=(7/2)lim(x->0+) tan(2x)/tan(7x)
=(7/2)(2/7)
=1