求lim x→0 (1+3x)5x分之一的平方,极限

题目

求lim x→0 (1+3x)5x分之一的平方,极限

答案

方法1
lim (x→0)(1+3x)^(1/(5x))
ln(1+3x)^(1/(5x))
=1/(5x)*ln(1+3x)
=[ln(1+3x)]/(5x)
∴lim(x-->0)ln(1+3x)^(1/(5x))
=lim(x-->0)[ln(1+3x)]/(5x)
=lim(x-->0)[3/(1+2x)]/5
=3/5
∴lim (x→0)(1+3x)^(1/(5x))=e^(3/5)
方法2
设3x=1/t,x=1/(3t) 1/(5x)=3/5t
lim (x→0)(1+3x)^(1/(5x))
=lim(t-->∞) (1+1/t)^(3/5t)
=lim(t-->∞) [(1+1/t)^t]^(3/5)
=e^(3/5)