Question

数列极限

N->∞ 【tan(pi/4+2/N)】exp N
tan(π/4+2/n)外面还有N次方呢

Answer 1

令y=[tan(π/4+2/N)]^N
ln(y)=Nln[tan(π/4+2/N)]
再令x=1/N。当N→∞，x→0。代入上式得：
ln(y)=(x→0)lim[ln(tan(π/4+2x))]/x
=(x→0)lim(2/tan(π/4+2x)=2
所以：
当N→∞时，y=[tan(π/4+2/N)]^N=e²

Answer 2

tan(π/4+2/n)=(tan(π/4)+tan(2/n))/(1-tan(π/4)*tan(2/n))
=(1+tan(2/n))/(1-tan(2/n))

n趋向于正无穷时，tan(2/n)趋向于0，tan(π/4+2/n)趋向于1