求n→∞时,(3n+2)^(1/n)的极限

求n→∞时,(3n+2)^(1/n)的极限
求n→∞时,(3n+2)^(1/n)的极限

求n→∞时,(3n+2)^(1/n)的极限
当n->∞时,lim(3n+2)^(1/n) = e^lim[(1/n)ln(3n+2)]
根据洛比达法则：n->∞时,lim[(1/n)ln(3n+2)] = 3/(3n+2) = 0
故：当n->∞时,lim(3n+2)^(1/n) = e^lim[(1/n)ln(3n+2)] = e^0 = 1

原式=
n→∞时，lim(n)^(1/n)*(3+2/n)^(1/n)
=limn^(1/n)*lim(3+2/n)^(1/n)
=1
PS:
1设其为1+a
(1+a)^n=n
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1+na+a/2*n*(n-1)a趋近于0当趋近于无穷

(3n+2)^(1/n)=[1+(3n+1)]^{[1/(3n+1)]*(1/n)*(3n+1)}
因为[1+(3n+1)]^{[1/(3n+1)]的极限等于e，而)(1/n)*(3n+1)=(3n+1)/n的极限等于3.
而[1+(3n+1)]^{[1/(3n+1)]*(1/n)*(3n+1)}
=e^ln{[1+(3n+1)]^{[1/(3n+1)]*(1/n)*(3n+1)}}
=e^{[(3n+1)/n]*{ln{[1+(3n+1)]^{[1/(3n+1)]}}}
=e^[3*lne]=e^3