Download 2.5 Continuity the Intermediate Value Theorem and the bisection

Example 2.4.2. Determine if f (x) = | − x + 2| has a limit at x = 2, if it has evaluate
the limit. Answer the same question for
�
x2 if x > 0

x − 1 if x < 0
g(x) =

2 if x = 0.
at x = 0.
Infinite limits and vertical asymptotes
Definition 2.4.2. Let f be defined on some open interval (a� c).
1. If for all N > 0 there exists δ > 0 such that if
a < x < a + δ then f (x) > N�
we say that the limit of f as x approaches a from the right is ∞ and we write
limx→a+ f (x) = ∞.
2. If for all N < 0 there exists δ > 0 such that if
a < x < a + δ then f (x) < N�
we say that the limit of f as x approaches a from the right is −∞ and we write
limx→a+ f (x) = −∞.
3. In either of the above two cases, we say that the line x = a is a vertical
asymptote of the graph of f , and that f has an infinite right-hand limit at a.
4. We can define in the same way the infinite left-hand limit of f at a.
Example 2.4.3. limx→0+ x1 = ∞� limx→0− x1 = −∞� limx→0+ ln x = −∞.
2
.
Find all the vertical asymptotes of the graph of f (x) = xx2 −1
−4
�a)
= ±∞, then
Definition 2.4.3. Suppose that f is continuous at a. If limx→a f �x)−f
x−a
we say that the graph of f has a vertical tangent line at (a� f (a)). In this case the
vertical line x = a is called the tangent to the graph of f at a.
Example 2.4.4. Find the tangent line to the graph of f (x) = x1�3 at (0� f (0)).
2.5
Continuity� the Intermediate Value Theorem�
and the bisection method
Definition 2.5.1. Let f be defined on some open interval containing a number a
including a itself. f is continuous at a if limx→a f (x) exists and limx→a f (x) = f (a).
If f is not continuous at a we say that it is discontinuous at a.
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Theorem 2.5.1. Suppose f and g are continuous at a and let c be any number.
Then, the functions f + g� cf� f g are continuous at a. In addition, if g(a) �= 0, then
f
is continuous at a.
g
If f is continuous at a and g is continuous at f (a), then g ◦ f is continuous at a.
Example 2.5.1. Determine for which values of x the following functions are continuous:
x
f (x) = x sinx2x+ln
, and g(x) = ln cos x.
+c
Definition 2.5.2. A function f is continuous from the right at a number a in its
domain, if limx→a+ f (x) = f (a).
f is continuous from the left at a number a in its domain, if limx→a− f (x) = f (a).
Remark 2.5.1. f is continuous at a number a in its domain if and only if it is continuous from the right and from the left at a.
Example 2.5.2.
g(x) =
�
x2 + 2x if x ≥ 0
sin x if x < 0.
Is g continuous at x = 0?
Definition 2.5.3.
1. f is continuous on an open interval (a� b) if it is continuous at
every number in the interval.
2. f is continuous on a closed interval [a� b] if it is continuous at every number in
the open interval (a� b) and continuous from the right at a and from the left at
b.
Example 2.5.3. For which real
√ numbers is each ofx the following functions continuous?
f (x) = ln(x + 2)� g(x) = 8 − x2 � h(x) = sin
.
�x�
Theorem 2.5.2 (Intermediate value theorem (IVT)). Let f be continuous on a closed
interval [a� b]. Let p be any number between f (a) and f (b) so that f (a) ≤ p ≤ f (b)
or f (b) ≤ p ≤ f (a). Then there exists a number c in [a� b] such that f (c) = p.
Example 2.5.4. Determine the intervals on which f is positive, and the intervals on
which it is negative where f (x) = (x2 − 1)(x + 2)(−2x + 3).
Example 2.5.5. Prove that the equation x3 + x + 1 = 0 has at least one zero. Use the
bisection method to find an approximation of this zero with an error less than 10−3 .
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