题目

lim x趋向0[e^(x^2)-e^(2-2cosx)]/x^4 答案是1/12

提问时间：2021-03-04

答案

lim(x→0) [e^(x^2)-e^(2-2cosx)]/x^4
=lim(x→0) [e^(x^2)-e^2/e^(2cosx)]/x^4
=lim(x→0) [e^(x^2)*e^(2cosx)-e^2]/[e^(2cosx)*x^4]
=lim(x→0) [e^(x^2)*e^(2cosx)-e^2]/[e^2*x^4]
=lim(x→0) [e^(x^2+2cosx)-e^2]/[e^2*x^4] (0/0)
=lim(x→0) [e^(x^2+2cosx)*(2x-2sinx)]/[4e^2*x^3]
=lim(x→0) [e^2*(2x-2sinx)]/[4e^2*x^3]
=lim(x→0) (x-sinx)/[2x^3] (0/0)
=lim(x→0) (1-cosx)/(6x^2)
=lim(x→0) (x^2/2)/(6x^2)
=1/12