7
: .
.
a(x) b(x) 0 (¥)
1) a(x) ~ b(x) 0 (¥) lim a(x)/b(x)=1 xx0 (¥)
2) a(x) b(x) 0 (¥) lim a(x)/b(x)=¹0 xx0 (¥)
3) a(x) b(x) 0 (¥) lim a(x)/b(x)=0 xx0 (¥)
.
f(x) g(x) 0 (¥)
1) f(x) ~ g(x) 0 (¥) lim f(x)/g(x)=1 xx0 (¥)
2)f (x) g (x) , 0 (¥) limf(x)/g(x)=¹¥ xx0 (¥) <¥
, =1,
1) f (x) g (x) g(x) g(x) 0 (¥) lim f (x)/g (x)=0 xx0 (¥)
:
1) sin(x)
x 0
lim sin(x)/x=1 Û sin(x)~x
x0
0
2) 1n(1+x)
0
lim ln(1+x)/x= lim ln(1+x)1/x =1 Û
x0 x0
ln(1+x) ~ x, 0
3) x2
22+1, +¥
lim x2/(2x2+1) = lim x2/x2(2+1/x2)=1/2
x+¥ x+¥
. :
const, .
:
1) a()=b() 0(¥). , a() b() 0 (¥), a()=g()b(), 0 (¥). - a() , b() a()/b()=g() , lim a(x)/b(x)=0 x0 (¥)
2) fa()=gb() 0(¥). , fa() g () 0 (¥), f ()=g()g (). - f () , g() f()/g ()=g() , lim f (x)/g (x)=0 x0 (¥)
.
.
xn=o(xm), 0<n<m +¥. , .
:
xn=xm(xn/xm)=xm(1/x(m-n))=xmg(x) m-n>0 xmg(x)ºo(xm)
.
=(b), 1<a<b x+¥. , .
ax=ax(bx/bx)=ax(a/b)x=bxg(xºo(bx) (0<a/b<1)
ln(x)=o(xa), "a>0. .
ln(x)<x "x
lim ln(x)/xa=lim [(ln(x)/(xa/2xa/2))((a/2)/(a/2))]=
|
|
x0 x0
lim [(ln(x)/xa/2)(2/(axa/2)]
x0
.
lim (ln(x)/xa)=0 Û (lim(x))/xa=g(x) Û lna=xag(x)ºoxa,
x0
x+¥
.
Xk=o(ax), " k>0,a>1 x+¥ lim(xk)/(ax)=0
x+¥
: a(x) ~ a1(x) xx0 (¥)
b(x) ~ b1(x) xx0 (¥)
lim a(x)/b(x)=lim a1(x)/b1(x)
xx0 (¥) xx0 (¥)
:
lima(x)/b(x)=lim[a(x)a1(x)b1(x)]/[a1(x)b1(x)b(x)]=lim(a(x)/b(x))lim(b1(x)/b(x))lim(a1(x)/b1(x))=lim a1(x)/b1(x)
x0 x0 x0 x0 x0 x0
. : .
:
lim sin(x)/3x=limx/3x=1/3
x0 x0
: ( )
a1(x)+a2(x)++an(x), xx0 (¥)
.
a1(x) , a2()=(a1()),,an(x)=o(a1(x)) xx0 (¥)
:
a1(x)+a2(x)++an(x) ~ a1(x), xx0 (¥) a1() .
:
lim [a1(x)+a2(x)++an(x)]/a1(x)=lim[a1(x)+a1(x)g(x)++a1(x)g(x)]/a1(x)=lim[a1(x)(1+g1(x)++gn(x))]/a1(x)=1 xx0 (¥) xx0 (¥) xx0 (¥)
:
lim (ex+3x100+ln3x)/(2x+1000x3+10000=lim ex/2x=lim ex/(exg(x))=+¥
x+¥ x+¥ x+¥
2x=o(ex)ºexg(x)
.
ex-1 0. lim (ex-1)/x=1, ex-1 ~ x x0
x0
1-cosx 0. lim (1-cos x)/(x2/2)=lim{2sin(2x/2)]/[x2/2]=lim [2(x/2)2]/[x2/2]=1,
1-cos(x) ~ x2/2 0 (1+x)p-1 ~ px 0