高数第一节x趋向无穷时,lim(1+1/x+1/x^2)^x=e?难道是1/x^2可以无视掉么?为什么

Question

高数第一节
x趋向无穷时,lim(1+1/x+1/x^2)^x=e?
难道是 1/x^2可以无视掉么?为什么

Answer 1

lim(x->inf) (1 + 1/x + 1/x²)^x
= lim [1 + (x+1)/x²]^x
= lim { 1 + 1 / [x²/(x+1)] }^x
= lim {1 + 1 / [x²/(x+1)] }^[x²/(x+1) * (x+1)/x² * x],应用重要极限
= e^lim (x+1)/x
= e^lim (1+1/x)
= e^(1+0)
= e