知识大全 f(x)=(x^2+x-2)/(x^3-x^2+x-1)求limx→-1

Posted 2022-07-11 知

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f(x)=(x^2+x-2)/(x^3-x^2+x-1)求limx→-1

f（x）=[（x-1）（x+2）]/[（x-1）（x^2+1）],
limx→-1时limf(x)=lim(x+2)/（x^2+1）=1/2

limx趋近于1(x^2-3x+2)/(x^3-x^2+x-1)

lim<x→1>(x²-3x+2)/(x³-x²+x-1)
=lim<x→1>[(x-1)(x-2)]/[x²(x-1)+(x-1)]
=lim<x→1>[(x-1)(x-2)]/[(x-1)(x²+1)]
=lim<x→1>(x-2)/(x²+1)
=-1/2

求lim√(3-x)-√(1+x)/(x^2+x-2) (x->1)

解：
lim(x→1)[√(3-x)-√(1+x)]/(x^2+x-2)
=lim(x→1)[√(3-x)-√(1+x)])[√(3-x)+√(1+x)]/(x^2+x-2)[√(3-x)+√(1+x)]
=lim(x→1)[(3-x)-(1+x)]/(x^2+x-2)[√(3-x)+√(1+x)]
=lim(x→1) 2(1-x)/(x-1)(x+2)[√(3-x)+√(1+x)]
=- lim(x→1) 2/(x+2)[√(3-x)+√(1+x)]
=-2/(3×2√2)
=-√2/6

x^2+x-1=0 求x(1-(2/3-x))/(x+1)

x^2+x-1=0, x1=(√5-1)/2, x2=(-√5-1)/2. 且 x^2=1-x.
f(x)=x[1-2/(3-x)]/(1+x) = x(1-x)/[(3-x)(1+x)]
= (x-x^2)/(3+2x-x^2) = (2x-1)/(3x+2)
= (2x+4/3-7/3)/(3x+2) = 2/3-(7/3)/(3x+2)
则 f(x1) = 2/3-(7/3)(2/(3√5+1)= 2/3-(7/66)(3√5-1)=(17-7√5)/22.
f(x2) = 2/3-(7/3)(2/(-3√5+1)= 2/3+(7/66)(3√5+1)=(17+7√5)/22.
得所求的值是 (17±7√5)/22.

已知x^3-x^2+x-2=0，求x^4+2x^3-2x^2+x-1的值

x^4+2x^3-2x^2+x-1
=(x^4-x^3+x^2-2x)+3x^3-3x^2+3x-1
=x(x^3-x^2+x-2)+3(x^3-x^2+x)-1
=0+6-1
=5

求极限limx→1 sin(x^2-1)/x^2+x-2

解关于X的不等式 ax2+(a-2)x-2 > 0 其中引数a包含于R
ax2+(a-2)x-2 > 0
(ax-2)(x+1)>0
a[x-(2/a)](x+1) > 0
a>0时，[x-(2/a）](x+1)>0
如果2/a>=1,即0<a2/a且 x>-1
即 x>2/a
a=0时，原式变为-2x-2>0 x<-1
a<0时，[x-(2/a）](x+1)<0
x<2/a 或 x<-1

计算x/(x^2-1) + (x^-x+1)/(x^3-x^2+x-1)+(x^2+x+1)/(x^3+x^2+x+1)-2x^3/(x^4-1)

x/(x^2-1) + (x^2-x+1)/(x^3-x^2+x-1)+(x^2+x+1)/(x^3+x^2+x+1)-2x^3/(x^4-1)
=x/(x^2-1) -2x^3/(x^4-1) + (x^2-x+1)/(x^3-x^2+x-1)+(x^2+x+1)/(x^3+x^2+x+1)
=x*(x^2+1) /(x^4-1) -2x^3/(x^4-1)+ (x^2-x+1)/[(x-1)*(x^2 +1)]+ (x^2+x+1)/[(x+1)*(x^2+1)]
=(x^3-2x^3+x) / (x^4-1) + [(x^2-x+1)*(x+1)+ (x^2+x+1)*(x-1)] /[(x-1)*(x+1)*(x^2+1)]
=(-x^3+x) / (x^4-1) + 2x^3/(x^4 -1)
=(x^3+x) / (x^4-1)
=x*(x^2+1) / (x^4-1)
= x/(x^2-1)

求(x^2+x-1)/(x^2+x-2)的100阶导数

y=(x^2+x-1)/(x^2+x-2)=1+1/(x^2+x-2)=1+(1/(x-1)-1/(x+2))/3
1/(x-1)的100阶导数100！（x-1）^-101
1/(x+2) 的100阶导数为100！（x+2）^-101
将上式代入得：
100！[（x-1）^-101 -- (x+2）^-101]/3

约分 (x^3-x^2+x-1)/(x-1)=

x^3-x^2+x-1化为x^2(x-1)+(x-1)
提公因式（x-1）(x^2+1)
所以最后约去（x-1）原式=x^2+1

求极限limx→1 sin(x^2-1)/x^2+x-2 急

limx→1 sin(x^2-1)/x^2+x-2
=(x^2-1)/(x^2+x-2)
=(x+1)/(x+2)(x->1)
=2/3