一道高数题，如图，求这个极限的解题过程，谢谢

Question

如题所述

Answer 1

望采纳！谢谢😜

本回答被提问者采纳

Answer 2

追问

很棒，谢谢您

追答

不用谢。

Answer 3

Answer 4

lim『x→∞』((x²+1)/(x-2))e^(1/x) - x
=lim『x→∞』x(((x²+1)/x(x-2))e^(1/x) - 1)
=lim『x→∞』x(((x²+1)/(x²-2x))e^(1/x) - 1)
=lim『x→∞』(((1+1/x²)/(1-2/x))e^(1/x) - 1)/(1/x)
=lim『1/x →0』(((1+1/x²)/(1-2/x))e^(1/x) - 1)/(1/x)
=lim『u →0』(((1+u²)/(1-2u))e^u - 1)/u
=lim『u →0』(2u(1-2u)+2(1+u²))/(1-2u)² e^u
+ ((1+u²)/(1-2u)) e^u
=lim『u →0』((2u+2-2u²)/(1-2u)²+(1+u²)(1+2u)/(1-2u)²) e^u
=lim『u →0』((2u³-u²+4u+3)/(1-2u)²) e^u
=3