x趋向于无穷时,（x-2/x+1）的2x+3次方的极限

题目

x趋向于无穷时,（x-2/x+1）的2x+3次方的极限
如题

答案

原式=limx→∞ {[1-3/(x+1)]^[-(x+1)/3]}^[-3(2x+3)/(x+1)]
=e^limx→∞ -3(2x+3)/(x+1)
=e^limx→∞ -3(2+3/x)/(1+1/x)
=e^[-3*(2+0)/(1+0)]
=e^-6.