шамин с сдо
.pdf
√
bn = n.
M > 0 |
N(M) = M2 |
√
bn = n > M,
n > N(M)
∞
{an}
{ank }
lim ank = ∞.
k→∞
{an} |
M > 0 |
n = n(M) an(M) > M.
nk
n1 = 1, nk > nk−1, ank > k.
{ank }
{an}
lim an = lim sup an.
n→∞ n→∞
{an}
lim an = lim inf an.
n→∞ n→∞
∞
−∞
an = (−1)n,
lim an = 1, lim an = −1.
n→∞ n→∞
f(x)
x0
x0
f(x) x → x0
f(x)
x → x0 |
A |
ε > 0 |
δ = δ(ε) >
|f(x) − A| < ε,
x =6 0
|x − x0| < δ.
f(x)
x → x0 |
A |
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xn |
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f(x) |
xn → x0 |
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f(xn) → A, |
n → ∞. |
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A |
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f(x) |
x → x0 |
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{xn} |
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x0 |
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xn 6= x0 |
xn |
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f(x) |
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f(xn) |
A |
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ε > 0 |
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A |
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δ = δ(ε) > 0 |
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|f(x) − A| < ε |
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0 < |x − x0| < δ |
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xn x0 |
δ > 0 |
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N = N(δ) |
n > N(δ) |
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|xn − x0| < δ.
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n > N(δ) |
|f(xn) − A| < ε. |
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{f(xn)} |
A |
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f(x) x → |
x0 |
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M > 0 |
δ > 0 |
x |
f(x) |
0 < |x − x0| < δ |
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|f(x) − A| > M. |
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δn = 1 |
δn |
xn |
n |
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1
|xn − x0| < δn = n
|f(x) − A| > M.
xn → x0
{f(xn)} |
A |
f(x) =
x2 + 6x − 7
x2 + 2x − 3
x0 = 1
x0 = 1 |
U1 = (0, 1) (1, 2) |
lim = 2.
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x→1 |
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x U1 |
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|f(x) − 2| = |
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x2 + 6x |
− |
7 |
− 2 |
= |
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(x |
− |
1)(x + 7) |
− 2 |
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x2 |
+ 2x |
3 |
(x |
1)(x + 3) |
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(x + 7) |
− |
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(x + 3) |
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(x + 3) |
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2 = |
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(1 |
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x) |
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ε > 0 |
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|1 − x| < ε. x + 3
x U1
|1 − x| < ε.
δ(ε) = ε
lim = 2
x→1
{an}
1 |
U1 |
fn = f(an)
fn → 2 n → ∞ |
an U1 |
f(an) = an + 7 an + 3
lim fn = 2.
n→∞
f(x) = sin x1 .
x → 0
αn = |
1 |
, βn = |
1 |
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π/2 + 2πn |
3/2π + 2πn |
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αn → 0 βn → 0 |
n → ∞ |
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f(αn) = sin(π/2 + 2πn) = 1,
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f(βn) = sin(3/2π + 2πn) = −1. |
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f(an) |
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{an} |
0 |
f(x) |
x → 0 |
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x → x0 |
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x x0 |
x |
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x0 |
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x |
x0 |
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f(x) |
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(a, b) |
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f(x) |
x → x0 |
1.00 |
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0.75 |
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0.50 |
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0.25 |
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1 sinx |
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0.00 |
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−0.25 |
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−0.50 |
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−0.75 |
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−1.00 |
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−4 |
−2 |
0 |
2 |
4 |
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x |
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sin 1 |
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n |
x0 (a, b] |
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A |
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ε > 0 |
δ = δ(ε) |
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|f(x) − A| < ε,
x0 − δ < x < x0.
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f(x) |
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(a, b) |
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f(x) x → x0 |
x0 [a, b) |
A |
ε > 0 |
δ = δ(ε) |
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|f(x) − A| < ε, |
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x0 < x < x0 + δ.
lim f(x) = A,
x→x0−0
lim f(x) = A.
x→x0+0
sign(x) |
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xlim0 sign(x) = −1, |
lim sign(x) |
= 1. |
x +0 |
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→ |
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f(x) = x · ln x, |
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(0, ∞) |
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lim f(x) = 0. |
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x→+0 |
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x |
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f(x) |
(a, ∞) |
A |
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x → +∞ |
ε > 0 |
M > 0 |
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x > M |
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