10个个极限的运算求解1）lim（x-1）（3次根号下x-1）/（根号x-1)2）lim（x-∞）（x^2+1）^50/(x+1)^1003)lim(x-∞)2x+3/x+3次根号下X4)lim(x-0)sin4x/x5)lim(x-0)1-cos2x/xsinx6)lim(x-∞)[(x-1)/(x+3)]^(x+2)7)lim(x-∞)n[ln(n+3)-l
更新时间： 2025-08-27 00:43:48
1人问答
问题描述：
10个个极限的运算求解
1）lim（x-1）（3次根号下x-1）/（根号x-1)
2）lim（x-∞）（x^2+1）^50/(x+1)^100
3)lim(x-∞)2x+3/x+3次根号下X
4)lim(x-0)sin4x/x
5)lim(x-0)1-cos2x/xsinx
6)lim(x-∞)[(x-1)/(x+3)]^(x+2)
7)lim(x-∞)n[ln(n+3)-ln(n)]
8)lim(x-0)ln(1+x)/x
9)lim(x-0)(sin3x)^2/√(1+xsinx)-cosx
10)lim(x-∞)5^[2x/(x+3)]
郭波回答：
1）lim（x-1）（3次根号下x-1）/（根号x-1)
=lim（x-1）（3次根号下x-1）/（3次根号x-1)(3次根号x^2+3次根号x+1)
=lim（x-1）1/(3次根号x^2+3次根号x+1)
=1/3
2）lim（x-∞）（x^2+1）^50/(x+1)^100
=lim（x-∞）x^100+.../(x^100+...)
=1
3)lim(x-∞)2x+3/x+3次根号下X
=lim(x-∞)(2+3/x)/(1+1/3次根号下X^2)
=2
4)lim(x-0)sin4x/x
=lim(x-0)4*sin4x/4x
=lim(x-0)4*4x/4x
=4
5)lim(x-0)1-cos2x/xsinx
=lim(x-0)2(sinx)^2/xsinx
=lim(x-0)2sinx/x
=lim(x-0)2x/x
=2
6)lim(x-∞)[(x-1)/(x+3)]^(x+2)
=lim(x-∞)[1-4/(x+3)]^(x+2)
=lim(x-∞)[1-4/(x+3)]^((x+3)/4))(x+2)*4/(x+3)
=lim(x-∞)e^(x+2)*4/(x+3)
=lim(x-∞)e^(1+2/x)*4/(1+3/x)
=e^4
7)lim(x-∞)n[ln(n+3)-ln(n)]
=lim(x-∞)n[ln(n+3)/n]
=lim(x-∞)[ln(1+3/n)^n]
=lim(x-∞)[ln(1+3/n)^n/3*3]
=e^3
8)lim(x-0)ln(1+x)/x
=lim(x-0)x/x
=1
9)lim(x-0)(sin3x)^2/√(1+xsinx)-cosx
=lim(x-0)(sin3x)^2/(1+1/2xsinx)-cosx
=lim(x-0)(sin3x)^2/(1/2xsinx+2(sinx/2)^2)
=lim(x-0)(3x)^2/(1/2x^2+2*(x/2)^2)
=9/(1/2+1/2)
=9
10)lim(x-∞)5^[2x/(x+3)]
=lim(x-∞)5^[2/(1+3/x)]
=lim(x-∞)5^2
=25
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