Question

高数极限，求解答

Answer 1

lim(x→0) [e^(x²)-e^(2-2cosx)]/x^4
=lim(x→0) [e^(x²)-e²/e^(2cosx)]/x^4
=lim(x→0) [e^(x²)*e^(2cosx)-e²]/[e^(2cosx)*x^4]
=lim(x→0) [e^(x²)*e^(2cosx)-e²]/[e²*x^4]
=lim(x→0) [e^(x²+2cosx)-e²]/[e²*x^4] (0/0)
洛必达法则
=lim(x→0) [e^(x²+2cosx)*(2x-2sinx)]/[4e²*x³]
=lim(x→0) [e²*(2x-2sinx)]/[4e²*x³]
=lim(x→0) (x-sinx)/[2x³] (0/0)
=lim(x→0) (1-cosx)/(6x²)
=lim(x→0) (x²/2)/(6x²)
=1/12

追问

谢谢