.
1. f (x) 0 .
: (1)
a≠b. (2)
{ n} 0 ( n≠ 0),
, .. { f (n)} .
2. f (x) , .
. , . U1=(0-ε; 0+ε), ε>0. f (x) 1Î U1, │ f (1)│>1. U2=(0-ε/2; 0+ε/2), ε>0 , 2Î U2, │ f (2)│>2. , Un=(0-ε/n; 0+ε/n), f (n) > n, n → 0; f (n)→∞. .
3. 0 f (x) ≥b, . ( ).
4. 0 f (x)≥g(x), .
5. 0 f (x)≥g(x)≥h(x) f (x) h(x) → 0
.
.
. f (x) 0 , 0 {n}, n>0, {f(n)} .
:
.
= = f (x),
=+b = f (x) →+∞, f (x) f (x)= +b+α(),
. = f (x) →+∞ , ,
→-∞.
.
y=f(x) () . R1, 1, 2, , 1<2 -:
f(x1)<f(x2) (f(x1) >f(x2))
y=f(x) () . R1, 1, 2 1<2 -:
f(x1)≤f(x2) (f(x1) ≥f(x2))
, , - . .
|
|
.
.
1) lim f(x)sinx/x =1( →0) .
-. .. - y= sinx , , →0 1.
T
M
tgx
x
K A
O
MK= sinx , sinx<x<tgx,
1<x/ sinx<1/cosx
1>sinx/x>cosx
→0 lim cosx=1, lim 1=1. .
2) lim (1+1/x)x =e(→+ (-)∞) .
-.
1) +∞. . n, -:
n ≤ x< n+1 (1)
, >1,n>0. , : 1+1/ n ≥ 1+1/x> 1+1/(n+1)
(1), : (1+1/ n)n+1≥ (1+1/x)x> (1+1/(n+1))n f(x) ≥(1+1/x)x>g(x). →+∞,n →+∞, f(x) g(x)→. - - lim (1+1/x)x →( →+∞), ..
2) -∞. =-t, t>0.
(1+1/x)x=(1-1/t)-t =((t-1)/t)-t =(t/(t-1))t =(1+1/(t-1))t =(1+1/(t-1))t-1 (1+1/(t-1))x →*1= →-∞, .. t →+∞, ..