求极限的一道题n趋近于无穷limn^3(a^(1/n)-a^sin(1/n))(a>0)

Question

求极限的一道题
n趋近于无穷
lim n^3(a^(1/n) -a^sin(1/n)) (a>0)

Answer 1

a^(1/n) -a^sin(1/n) =e^(1/n)lna -e^sin(1/n)lna=1+i/nlna+1/(2n^2)(lna)^2+1/(6n^3)(lna)^3+o(1/n^3)-(1+sin(i/n)lna+1/2sin(1/n)^2(lna)^2+1/6sin(1/n)^3(lna)^3+o(1/n^3))=1+i/nlna+1/(2n^2)(lna)^2+1/(6n^3)(lna)^3-(1+(i/n-1/(6n^3)+o(1/n^3))lna+1/2(1+(i/n-1/(6n^3)+o(1/n^3))^2(lna)^2+1/6(1+(i/n-1/(6n^3)+o(1/n^3))^3(lna)^3+o(1/n^3))=1/(6n^3)lna+o(1/n^3)
所以lim n^3(a^(1/n) -a^sin(1/n))=lim n^3(1/(6n^3)lna+o(1/n^3))=1/6