lim根号n[根号(n+4)-根号(n-3)]n=>无穷

Question

lim根号n[根号(n+4)-根号(n-3)] n=>无穷

Answer 1

lim sqrt(n) [sqrt(n+4)-sqrt(n-3)]
= lim sqrt(n) [(n+4)-(n-3)] / [sqrt(n+4)+sqrt(n-3)]
= lim sqrt(n) [7 / [sqrt(n+4)+sqrt(n-3)]
= lim 7 /{ [sqrt(n+4)+sqrt(n-3)] /sqrt(n) }
= lim 7 /[sqrt(n+4) /sqrt(n)+sqrt(n-3) /sqrt(n)]
= 7 / [lim sqrt(1+4/n) + sqrt(1-3/n)]
=7 / (1+1)
=7/2 (n->无穷)
这里的sqrt(...)表示某数的开方
运用的技巧是"分子有理化"