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(n→∞)lim(3n%1/n)^(1/n)

(n→∞) lim(3n-1/n)^(1/n) =lim3^(1/n)*(n-1/3n)^(3/3n) =1/e^3

如图中::

如下图片:

n→∞时((n+1)/(n-1))的3n+2次方 =n→∞((1+2/(n-1))的3n+2次方 =limn→∞((1+2/(n-1))的(n-1)/2次方的2(3n+2)/(n-1)次方 =[limn→∞((1+2/(n-1))的(n-1)/2次方]的lim(n->∞)2(3n+2)/(n-1)次方 =e的lim(n->∞)2(3n+2)/(n-1)次方 =...

lim1/n(1+1/2+1/3+......+1/n)=lim(ln(n+1)+欧拉常数r)/n=lim1/(n+1)(洛必达法则)=0

lim(n^3+n)/(n^4-3n^2+1) =lim(1/n+1/n^3)/(1-3/n^2+1/n^4) =(0+0)/(1-0+0) =0

由于1/(n^3+3n^2+n)

lim(n→∞)(3n^2-n+1)/(2+n^2) =lim(n→∞)(3n^2+6-6-n+1)/(2+n^2) =lim(n→∞)[3(n^2+2)-(5+n)]/(2+n^2) =lim(n→∞)[3-(5+n)/(2+n^2)] =3

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