n趋近于无穷大时,lim{[a^(1/n)+b^(1/n)]/2}^n,a>0,

Question

n趋近于无穷大时,lim{[a^(1/n)+b^(1/n)]/2}^n,a>0,

Answer 1

这是1^∞型极限,用重要极限lim (x→∞) [1+(1/x)]^x=e
lim (n→∞) {[a^(1/n)+b^(1/n)]/2}^n
=lim (n→∞) [a^(1/n)]^n*{[1+(b/a)^(1/n)]/2}^n
=a*lim (n→∞) {1+[(b/a)^(1/n)-1]/2}^n
=a*lim (n→∞) {1+[(b/a)^(1/n)-1]/2}^({2/[(b/a)^(1/n)-1]}*{n*[(b/a)^(1/n)-1]/2})
=a*e^lim (n→∞) {n*[(b/a)^(1/n)-1]/2}.重要极限lim (x→∞) [1+(1/x)]^x=e
=a*e^lim (t→0) [(b/a)^t-1]/(2t)].换元t=1/n
=a*e^lim (t→0) [(b/a)^t]ln(b/a)/2.L'Hospital法则
=a*e^[ln(b/a)/2]
=a√(b/a)
=√(ab)
希望你能看清楚