lim(x→∞)[(x^3+x^2+x^1+1)^(1/3) - x] 的极限.

来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/25 15:44:54
lim(x→∞)[(x^3+x^2+x^1+1)^(1/3)-x]的极限.lim(x→∞)[(x^3+x^2+x^1+1)^(1/3)-x]的极限.lim(x→∞)[(x^3+x^2+x^1+1)^(

lim(x→∞)[(x^3+x^2+x^1+1)^(1/3) - x] 的极限.
lim(x→∞)[(x^3+x^2+x^1+1)^(1/3) - x] 的极限.

lim(x→∞)[(x^3+x^2+x^1+1)^(1/3) - x] 的极限.
=1/3
A=(x^3+x^2+x^1+1)^(1/3) B= x首先分析分数同时乘以(A^2+B^2+AB)
然后上下同时除以X^2 然后比如x^2/x^3=0 之类的 结果是1/3

用等价无穷小代换
lim(x→∞)[(x^3+x^2+x^1+1)^(1/3) - x]  (上下同除x)
=lim(x→∞)[(1+1/x+1/x^2+1/x^3)^(1/3) - 1] /(1/x) (1/x=t,x→∞,t→0)
=lim(t→0)[(1+t+t^2+t^3)^(1/3) - 1] /t (用等价无穷小代换)
=lim(t→0)(1/3t)/t
=1/3

lim(x→∞)[(x^3+x^2+x^1+1)^(1/3) - x]
=lim(x→∞)[(x^3+x^2+x^1+1)^(1/3) - x][(x^3+x^2+x^1+1)^(2/3)+ (x^3+x^2+x^1+1)^(1/3)x+x^3]/[(x^3+x^2+x^1+1)^(2/3)+ (x^3+x^2+x^1+1)^(1/3)x+x^3]
= lim(x→∞)(x^2+...

全部展开

lim(x→∞)[(x^3+x^2+x^1+1)^(1/3) - x]
=lim(x→∞)[(x^3+x^2+x^1+1)^(1/3) - x][(x^3+x^2+x^1+1)^(2/3)+ (x^3+x^2+x^1+1)^(1/3)x+x^3]/[(x^3+x^2+x^1+1)^(2/3)+ (x^3+x^2+x^1+1)^(1/3)x+x^3]
= lim(x→∞)(x^2+x^1+1)/[(x^3+x^2+x^1+1)^(2/3)+ (x^3+x^2+x^1+1)^(1/3)x+x^3]
= lim(x→∞)(1+1/x+1/x^2)/[(1+1/x+1/x^2+1/x^3)^(2/3)+(1+1/x+1/x^2+1/x^3)^(1/3)+1]
=1/3

收起