Question

高数 求极限大神帮忙

Answer 1

解：(1)，∵x→0，(1+x)^α~1+αx，设x=1/t，x→∞时，t→0，
∴原式=lim(t→0)[(1+3t)^(1/3)-(1-2t)^(1/4)]/t=lim(t→0)[(1+3t/3)-(1-2t/4)]/t=3/2。
(2)，∵x→0，e^x~1+x+(1/2)x^2、cosx~1-(1/2)x^2+(1/4!)x^4、ln(1-x)~-x-(1/2)x^2-(1/3)x^3，
∴cosx-e^(-x^2/2)~1-(1/2)x^2+(1/4!)x^4-[1-(1/2)x^2+(1/8)x^4]=(-1/12)x^4、x+ln(1-x)~-(1/2)x^2-(1/3)x^3
∴原式=lim(x→0)[(-1/12)x^4]/[-(1/2)x^4-(1/3)x^5]=1/6。
供参考。