(3x^2-1)/(x+1)e^（2/x）-3x 当x趋于无穷时的极限

lim(x→∞)[(3x²-1)e^(2/x)/(x+1)]-3x
=lim(x→∞)[(3x²-1)e^(2/x)-3x²-3x]/(x+1)
=lim(x→∞)[3x²[e^(2/x)-1]-3x-e^(2/x)]/(x+1)
=lim(x→∞){3x[e^(2/x)-1]-3-[e^(2/x)]/x}/(1+1/x)
=lim(x→∞){3x·2/x-3-[e^(2/x)]/x}/(1+1/x)
=lim(x→∞){6-3-[e^(2/x)]/x}/(1+1/x)
=3
答案：3

考虑其导数的极限。上下同除以x²得：lim x->∞ [(x+1)e^(2/x)-3x]/(3x²-1)=limx->∞ [(1/x+1/x²)*(e^(2/x)) -3/x] /(3-1/x²) =((0+0)*1 -0)/(3-0)=0
所以原式极限=∞