Question

数学lim…求过程

Answer 1

lim n->∞ n/(e^n+3n)=无穷大/无穷大型，用罗必塔法则
lim n->∞ 1/(e^n+3)
由于当n->+∞时，极限为0；当n->-∞时，极限为1/3，所以本题极限值不存在。

lim n->∞ (1/n)^(1/n)=lin n->∞ e^[1/n*ln(1/n)]
而lin n->∞ 1/n*ln(1/n)=lin n->∞ -lnn/n =无穷大/无穷大型，用罗必塔法则
lin n->∞ -(1/n)/1=0
所以原极限=e^0=1

Answer 2

lim n/(eⁿ+3n)
n→∞
=lim 1/(eⁿ+3)
n→∞
=lim (1/eⁿ)/(1+ 3/eⁿ)
n→∞
=0/(1+0)
=0

lim ln[(1/n)^(1/n)]
n→∞
=lim (1/n)ln(1/n)
n→∞
=lim (1/n)(ln1 -lnn)
n→∞
=lim (ln1)/n- lim(1/n)lnn
n→∞ n→∞
=0 -lim(lnn)/n
n→∞
=-lim (1/n)/1
n→∞
=0
lim (1/n)^(1/n) =1
n→∞