lim(x→∞) [(2-x)/(3-x)]^(x+2)
=lim(x→∞) [(3-x-1)/(3-x)]^(x+2)
=lim(x→∞) [1-1/(3-x)]^(x+2),之后根據(jù)e的定義lim(x→∞) (1+1/x)^x,將原式湊成這形式
=lim(x→∞) [1-1/(3-x)]^[-(3-x)]*[-(x+2)/(3-x)],留意指數(shù)那是-(3-x),有負號的
=e^-lim(x→∞) (x+2)/(3-x),lim(x→∞) [1-1/(3-x)]^[-(3-x)]這個等于e,之后對指數(shù)部分求極限
=e^-lim(x→∞) (1+2/x)/(3/x-1),分子分母各除以x
=e^-(1+0)/(0-1)
=e