Question

求极限lim(x→∞)[(2x+5)/(2x+3)]^(x+1)

Answer 1

原式=lim [1+ 2/(2x+3)]^(x+1)
=lim [1+2/(2x+3)]^{[(2x+3)/2]*[2/(2x+3)]*(x+1)}
=e^lim [(2x+2)/(2x+3)]
=e^1
=e...................当x→∞时
主要利用重要极限lim (1+1/x)^x=e，当x→∞时

Answer 2

先对此函数取指数函数 取出来为无穷乘以零 化为零比零型 用洛必达法则进行计算 计算一次后进行合并 合并无穷比无穷 接着做三次无穷比无穷的洛必达法则的计算 就得到了最后的答案为无穷

Answer 3

令x+1=y
lim(x→∞)[(2x+5)/(2x+3)]^(x+1)
=lim(y→∞)[(2y+3)/(2y+1)]^y
=lim(y→∞)[1+2/(2y+1)]^y
=lim(y→∞)[1+1/(y+1/2)]^(y+1/2-1/2)
=e