THEOREM 20.4:
If
lim f (x) = L,
x→c
then there exists a neighborhood N (c)
of c, such that f
is bounded on
N (c). That is, there is a number M
such that
|f (x)| ≤ M
for all x ∈ D ∩ N (c).
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THEOREM 20.5:
(Arithmetic)
Let f, g : D → R and let c be an
accumulation point of D. If
lim f (x) = L
x→c
and
lim g(x) = M,
x→c
then
1. x→c
lim [f (x) + g(x)] = L + M ,
2. x→c
lim [f (x) − g(x)] = L − M ,
3. x→c
lim [f (x)g(x)] = LM ,
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4 x→c
lim [k f (x)] = kL, k constant,
f (x)
L
5 x→c
lim
=
g(x)
M
provided M 6= 0.
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THEOREM 20.6:
Theorem”)
Let
(“Pinching
f, g, h : D → R
and let c be an accumulation point
of D. Suppose that
f (x) ≤ g(x) ≤ h(x)
for all x ∈ D, x 6= c. If
lim f (x) = x→c
lim h(x) = L,
x→c
then
lim g(x) = L.
x→c
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Some basic limits:
1. x→c
lim k = k
for any constant k.
2. x→c
lim x = c.
3. x→c
lim |x| = |c|.
4. For any positive number c,
lim
x→c
√
√
x = c.
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5. x→c
lim p(x) = p(c) for any polynomial function p(x).
6. x→c
lim R(x) = R(c) for any rational
function R(x), provided R(c) 6= 0.
7. lim sin x = 0
x→0
8. lim cos x = 1
x→0
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THEOREM 20.7:
The following
are equivalent:
lim f (x) = L,
x→c
lim (f (x)−L) = 0,
x→c
lim f (c+h) = L,
h→0
lim |f (x)−L| = 0.
x→c
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9. x→c
lim sin x = sin c
10. x→c
lim cos x = cos c
21
THEOREM 20.8:
R
and let
c
Let f : D →
be an accumulation
point of D. If
lim f (x) = L > 0,
x→c
then there exists a deleted neighborhood
N ∗(c)
of
c
such that
f (x) > 0 for all x ∈ N ∗(c) ∩ D.
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One-sided limits:
Def.
Let f : D → R and let c
be an accumulation point of D. A
number L is the right-hand limit
of f at c if to each > 0 there
exists a δ > 0 such that
|f (x) − L| < whenever
x∈D
Notation:
and
c < x < c + δ.
lim f (x) = L.
x→c+
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A number M is the left-hand limit
of f at c if to each > 0 there
exists a δ > 0 such that
|f (x) − M | < whenever
x∈D
Notation:
and
c − δ < x < c.
lim f (x) = M.
x→c−
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THEOREM 20.9:
lim f (x) = L
x→c
if and only if each of the one-sided
limits
lim f (x)
x→c+
and
lim f (x)
x→c−
exists, and
lim f (x) = lim f (x) = L.
x→c+
x→c−
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