Chapter 2: Functions, Limits and...
Transcript of Chapter 2: Functions, Limits and...
Chapter 2: Functions, Limits and Continuity
• Functions
• Limits
• Continuity
Chapter 2: Functions, Limits and Continuity 1
Functions
Functions are the major tools for describing the real world inmathematical terms.
Definition 1. A function f is a set of ordered pairs of numbers(x, y) satisfying the property that if (x, y1), (x, y2) ∈ f theny1 = y2 (that is, no two distinct ordered pairs in f have thesame first component).
If f is a function from a set X into a set Y , then we write
f : X → Y read as “f is a function from set X into set Y .
If (x, y) ∈ f , then we may write
y = f(x) (read as “y equals f of x”)
Chapter 2: Functions, Limits and Continuity 2
so that y is called the image of x under the function f ; x is a pre-imageof y under f . We also say that “y is the function value of x under f .”
In the function f : X → Y , the set X containing all of the firstcomponents of ordered pairs in f is called the domain of the function f ;the set Y is called the co-domain of f . The set of all second componentsof ordered pairs in f is called that range of f , that is,
range f = {y ∈ Y : y = f(x) for some x ∈ X}.
Clearly,range f ⊆ Y.
Chapter 2: Functions, Limits and Continuity 3
Example 1. Let f = {(x, y) : y =√x− 2}.
• The value of y that corresponds to x = 6 is y =√6− 2 =
√4 = 2.
Hence, f(6) = 2 and (6, 2) ∈ f .
• The domain of f is the set
dom f = [2,+∞)
and the range of f is the set
range f = R≥0 = [0,∞).
Chapter 2: Functions, Limits and Continuity 4
Definition 2. If f is a function, then the graph of f is the setof all points (x, y) in a given plane for which (x, y) ∈ f .
Example 2. The graph of f(x) = x− 2 is given by
Chapter 2: Functions, Limits and Continuity 5
Example 3. The graph of g(x) = x2 − 4 is given by
Chapter 2: Functions, Limits and Continuity 6
Example 4. The graph of h(x) =√x2 − 9 is given by
Chapter 2: Functions, Limits and Continuity 7
Example 5. The graph of h(x) =√x(x− 2) is given by
Chapter 2: Functions, Limits and Continuity 8
Example 6. The graph of F (x) =
3x− 1, if x < 24, if x = 27− x, if x > 2.
is given by
Chapter 2: Functions, Limits and Continuity 9
Example 7. The graph of G(x) =
{4− x2, if x ≤ 210, otherwise.
is given by
Vertical Line Test
A vertical line intersects the graph of a function f in at mostone point.
Chapter 2: Functions, Limits and Continuity 10
Limits
Example 8. How does the function f(x) =2x2 + x− 3
x− 1behave near x = 1?
x f(x) =2x2 + x− 3
x− 10 0.30.25 3.50.5 40.75 4.80.9 4.980.99 4.9980.999 4.99980.9999 4.999980.99999 4.999998
x less than 1
x f(x) =2x2 + x− 3
x− 12 71.75 6.51.5 6.01.25 5.51.1 5.21.01 5.021.001 5.0021.0001 5.00021.00001 5.00002
x greater than 1
Chapter 2: Functions, Limits and Continuity 11
Definition 3. (Informal Definition of Limit) Let f(x) bedefined on an open interval containing a, except possibly at aitself. If f(x) gets arbitrarily close to L for all x sufficientlyclose to a, we say that f approaches the limit L as x approachesa, and we write
limx→a
f(x) = L.
Example 9. Let f be defined by
f(x) =
x2 − 1
x− 1, if x 6= 1
1, if x = 1
Chapter 2: Functions, Limits and Continuity 12
Example 10. Find the following limits (if they exist).
1. limx→4
2
2. limx→−2
x
3. limx→0
(5x− 3)
Chapter 2: Functions, Limits and Continuity 13
Remark 1. A function may fail to have a limit at a point in its domain.
Example 11. Discuss the behavior of the following functions as x approachesa = 0.
1. U(x) =
{0, x < 01, x ≥ 0
Chapter 2: Functions, Limits and Continuity 14
2. g(x) =
{1/x, x 6= 00, x = 0
Chapter 2: Functions, Limits and Continuity 15
3. f(x) =
{0, x ≤ 0sin(1/x), x > 0
Chapter 2: Functions, Limits and Continuity 16
Limits from Graphs
For the given function f(x) with its graph shown, find the followinglimits.
Example 12. 1. limx→1
f(x)
2. limx→2
f(x)
3. limx→3
f(x)
Chapter 2: Functions, Limits and Continuity 17
Example 13. 1. limx→−2
f(x)
2. limx→−1
f(x)
3. limx→0
f(x)
Chapter 2: Functions, Limits and Continuity 18
Example 14. 1. limx→2
f(x)
2. limx→1
f(x)
3. limx→0
f(x)
Chapter 2: Functions, Limits and Continuity 19
Definition 4. (Formal Definition of Limit) Let f be a functiondefined at every number in some open interval containing a,except possibly at the number a itself. The limit of f(x) as xapproaches a is L, written as
limx→a
f(x) = L
if the following statement is true:Given any ε > 0, however small, there exists a δ > 0 such that
if 0 < |x− a| < δ then |f(x)− L| < ε.
Chapter 2: Functions, Limits and Continuity 20
Example 15. Let f(x) = 2x− 5.
1. Find a δ > 0 such that whenever 0 < |x−3| < δ whenever |f(x)−1| < εwhere ε = 0.1.
2. Show that limx→3
f(x) = 1.
Chapter 2: Functions, Limits and Continuity 21
Example 16. Let f(x) = x2.
1. Find a δ > 0 such that whenever 0 < |x−2| < δ whenever |f(x)−4| < εwhere ε = 0.3.
2. Show that limx→2
f(x) = 4.
Chapter 2: Functions, Limits and Continuity 22
The Limit TheoremsTheorem 1. (Limit of a Linear Function)If m and b are constants, then
limx→a
mx+ b = ma+ b.
Theorem 2. (Limit of a Constant)If c is a constant, then for any number a
limx→a
c = c.
Theorem 3. (Limit of the Identity Function)
limx→a
x = a.
Chapter 2: Functions, Limits and Continuity 23
Theorem 4. (Limit of the Sum and Difference of Two Functions)If limx→a
f(x) = L and limx→a
g(x) =M , then
limx→a
[f(x)± g(x)] = L±M.
Theorem 5. (Limit of the Sum and Difference of n Functions)If limx→a
f1(x) = L1, limx→a
f2(x) = L2, . . ., limx→a
fn(x) = Ln, then
limx→a
[f1(x)± f2(x)± · · · ± fn(x)] = L1 ± L2 ± · · · ± Ln.
Theorem 6. (Limit of the Product of Two Functions)If limx→a
f(x) = L and limx→a
g(x) =M , then
limx→a
[f(x)g(x)] = LM.
Chapter 2: Functions, Limits and Continuity 24
Theorem 7. (Limit of the Product of n Functions)If limx→a
f1(x) = L1, limx→a
f2(x) = L2, . . ., limx→a
fn(x) = Ln, then
limx→a
[f1(x) · f2(x) · · · fn(x)] = L1 · L2 · · ·Ln.
Theorem 8. (Limit of the nth Power of a Function)If limx→a
f(x) = L and n is any positive integer, then
limx→a
[f(x)]n = Ln.
Theorem 9. (Limit of the Quotient of Two Functions)If limx→a
f(x) = L and limx→a
g(x) =M , then
limx→a
[f(x)/g(x)] = L/M if M 6= 0.
Chapter 2: Functions, Limits and Continuity 25
Theorem 10. (Limit of the nth Root of a Function)If limx→a
f(x) = L and n is any positive integer, then
limx→a
n√f(x) =
n√L
with the restriction that if n is even, L > 0.Theorem 11.
For any real number a except 0
limx→a
1
x=
1
a
with the restriction that if n is even, L > 0.
Chapter 2: Functions, Limits and Continuity 26
Theorem 12.
For a > 0 and n a positive integer, or if a ≤ 0 and n is an odd positiveinteger, then
limx→a
n√x = n
√a.
Theorem 13.
limx→a
f(x) = L if and only if limx→a
[f(x)− L] = 0.
Theorem 14.
limx→a
f(x) = L if and only if limt→0
f(t+ a)] = L.
Theorem 15.
limx→a
f(x) = L1 and limx→a
f(x) = L2 implies L1 = L2.
Chapter 2: Functions, Limits and Continuity 27
Exercises:Find the indicated limit .
1. limx→−4
(5x+ 2)
2. limx→3
(2x2 − 4x+ 5)
3. limy→−1
(y3 − 2y2 + 3y − 4)
4. limx→2
3x+ 4
8x− 1
5. limx→−1
2x+ 1
x2 − 3x+ 4
6. limx→2
√x2 + 3x+ 4
x3 + 1
7. limx→−3
3
√5 + 2x
5− x
8. limz→−5
z2 − 25
z + 5
9. limx→1/3
3x− 1
9x2 − 1
10. limx→−1
√x+ 5− 2
x+ 1
Chapter 2: Functions, Limits and Continuity 28
One-Sided Limits
Definition 5. (Definition of Right-Hand Limit) Let f be afunction defined at every number in some open interval (a, c).The limit of f(x) as x approaches a from the right is L, writtenas
limx→a+
f(x) = L
if for any ε > 0, however small, there exists a δ > 0 such thatif 0 < x− a < δ then |f(x)− L| < ε.
Chapter 2: Functions, Limits and Continuity 29
-1 0 5 0 0 -1 0 8 1 -.25 1 .25 4 -.25 4 .25x0 x0 + δ
x•1.8 4 1.8
δ
for all x 6= x0 in here
.5 -3 2.5 -1.5 1 -1.2 1 -1.3 4 -1.3 4 -1.2 /
L− ε
L
L+ ε
f(x) lies in here
.6 3 .8 3 .8 7 .6 7 / .6 5 1 5 -.1 3 .1 3 -.1 5 .1 5 -.1 7 .1 7
f(x)•
Chapter 2: Functions, Limits and Continuity 30
Definition 6. (Definition of Left-Hand Limit) Let f be afunction defined at every number in some open interval (d, a).The limit of f(x) as x approaches a from the left is L, writtenas
limx→a+
f(x) = L
if for any ε > 0, however small, there exists a δ > 0 such thatif 0 < a− x < δ then |f(x)− L| < ε.
Chapter 2: Functions, Limits and Continuity 31
-1 0 5 0 0 -1 0 8 1 -.25 1 .25 4 -.25 4 .25
x0 − δ x0
x•1.8 4 1.8
δ
for all x 6= x0 in here
.5 -3 2.5 -1.5 1 -1.2 1 -1.3 4 -1.3 4 -1.2 /
L− ε
L
L+ ε
f(x) lies in here
.6 3 .8 3 .8 7 .6 7 / .6 5 1 5 -.1 3 .1 3 -.1 5 .1 5 -.1 7 .1 7
f(x)•
Chapter 2: Functions, Limits and Continuity 32
Example 17. Let sgn(x) =
−1, if x < 00, if x = 01, if x > 0.
.
Find limx→0+
sgn(x), limx→0−
sgn(x) and limx→0
sgn(x).
Chapter 2: Functions, Limits and Continuity 33
Theorem 16. limx→a
f(x) exists and is equal to L if and only if limx→a+
f(x)
and limx→a−
f(x) both exist and both are equal to L.
Example 18. Let f(x) be the function defined by f(x) =|x|x
. Then
limx→0+|x|x
= 1 and limx→0−|x|x
= −1. Using Theorem (16), we see that
limx→0|x|x
does not exist.
Chapter 2: Functions, Limits and Continuity 34
Example 19. Let f be defined by
f(x) =
x+ 5, if x < −3√9− x2, if − 3 ≤ x ≤ 3
3− x, if 3 < x.
Chapter 2: Functions, Limits and Continuity 35
Exercises:Sketch the graph of the function and find the indicated limit if it exists; ifit does not exist, state the reason.
1. g(t) =
3 + t2, if t < −20, if t = −211− t2, if t > −2
(a) limt→−2+
g(t)
(b) limt→−2−
g(t)
(c) limt→−2
g(t)
Chapter 2: Functions, Limits and Continuity 36
2. f(x) =
2x+ 3, if x < 14, if x = 1x2 + 2, if x > 1
(a) limx→1+
f(x)
(b) limx→1−
f(x)
(c) limx→1
f(x)
3. f(x) =
x+ 1, if x < −1x2, if − 1 ≤ x ≤ 12− x, if x > 1
(a) limx→−1+
f(x)
(b) limx→−1−
f(x)
(c) limx→−1
f(x)
(d) limx→1+
f(x)
(e) limx→1−
f(x)
(f) limx→1
f(x)
Chapter 2: Functions, Limits and Continuity 37
More Exercise
Find each of the following for the given function and specified a value.
limx→a−
f(x), limx→a+
f(x), and limx→a
f(x)
1. f(x) =
2, if x < 1
−1, if x = 1
−3, if 1 < x.
;
a = 1
2. f(x) =
{x+ 4, if x ≤ −44− x, otherwise.
;
a = −4
3. f(x) =
{x2, if x ≤ 2
8− 2x, otherwise.;
a = 2
4. f(x) =
2x+ 3, if x < 1
2, if x = 1
7− 2x, if 1 < x.
;
a = 1
5. f(x) =
x2 − 4, if x < 2
4, if x = 2
4− x2, if 2 < x.
;
a = 2
6. f(x) = |x− 5|;a = 0
7. f(x) =
2, if x < −2√4− x2, if − 2 ≤ x ≤ 2
−2, if 2 < x.
;
a = −2 and a = 2
8. f(x) =
√x2 − 9, if x ≤ −3√9− x2, if − 3 < x < 3√x2 − 9, if 3 ≤ x.
;
a = −3 and a = 3
9. f(x) =
3√x+ 1, if x ≤ −1√1− x2, if − 1 < x < 1
3√x− 1, if 1 ≤ x.
;
a = −1 and a = 1
Chapter 2: Functions, Limits and Continuity 38
Infinite Limits
One-sided infinite limits
Example 20. Find limx→1+
1
x− 1and lim
x→1−
1
x− 1.
Chapter 2: Functions, Limits and Continuity 39
Two-sided infinite limits
Example 21. Find limx→0
1
x2.
Chapter 2: Functions, Limits and Continuity 40
Definition 7. Infinite Limits
1. Let f be a function defined at every number in some open interval
containing the number a except possibly at the number a itself.
As x approaches a, f(x) increases without bound, which is written
limx→a f(x) = +∞, if for any number N > 0 there exists a δ > 0
such that
if 0 < |x− a| < δ then f(x) > N.
2. Let f be a function defined at every number in some open interval
containing the number a except possibly at the number a itself. As
x approaches a, f(x) decreases without bound, which is written
limx→a f(x) = −∞, if for any number N < 0 there exists a δ > 0
such that
if 0 < |x− a| < δ then f(x) < N.
Chapter 2: Functions, Limits and Continuity 41
Example 22. f(x) =2x
x− 1
Chapter 2: Functions, Limits and Continuity 42
Theorem 17. If r is any positive integer, then
(i) limx→0+
1
xr= +∞
(ii) limx→0−
1
xr=
{−∞, if r is odd+∞, if r is even.
Example 23. Find
(a) limx→0+
1
x3
(b) limx→0−
1
x5
(c) limx→0−
1
x6
Chapter 2: Functions, Limits and Continuity 43
Theorem 18. If a is any real number and if limx→a f(x) = 0 and limx→a g(x) = c,
where c is a constant not equal to 0, then
(i) if c > 0 and if f(x)→ 0 through positive values of f(x), then
limx→a
g(x)
f(x)= +∞
(ii) if c > 0 and if f(x)→ 0 through negative values of f(x), then
limx→a
g(x)
f(x)= −∞
(iii) if c < 0 and if f(x)→ 0 through positive values of f(x), then
limx→a
g(x)
f(x)= −∞
(iv) if c < 0 and if f(x)→ 0 through negative values of f(x), then
limx→a
g(x)
f(x)= +∞
Chapter 2: Functions, Limits and Continuity 44
The theorem is also valid if “x→ a” is replaced by “x→ a+” or “x→ a−.”
Example 24. Find limx→1−
2x
x− 1and lim
x→1+
2x
x− 1
Example 25. Find limx→3+
x2 + x+ 2
x2 − 2x− 3and lim
x→3−
x2 + x+ 2
x2 − 2x− 3
Example 26. Let f(x) =
√x2 − 4
x− 2and g(x) =
√4− x2
x− 2.
Find (a) limx→2+
f(x), (b) limx→2−
g(x)
Chapter 2: Functions, Limits and Continuity 45
Example 27. Let F (x) =x2 + x+ 2
x2 − 2x− 3. Find
1. limx→3+ F (x)
2. limx→3− F (x)
3. limx→−1+ F (x)
4. limx→−1− F (x)
Chapter 2: Functions, Limits and Continuity 46
Theorem 19.
(i) If limx→a
f(x) = +∞ and limx→a
g(x) = c where c is any constant, then
limx→a
[f(x) + g(x)] = +∞.
(ii) If limx→a
f(x) = −∞ and limx→a
g(x) = c where c is any constant, then
limx→a
[f(x) + g(x)] = −∞.
The theorem holds if “x→ a” is replaced by “x→ a+” or “x→ a−.”
Chapter 2: Functions, Limits and Continuity 47
Theorem 20. If limx→a
f(x) = +∞ and limx→a
g(x) = c where c is any
constant except 0, then
(i) if c > 0, limx→a f(x) · g(x) = +∞;
(ii) if c < 0, limx→a f(x) · g(x) = −∞.
The theorem holds if “x→ a” is replaced by “x→ a+” or “x→ a−.”
Chapter 2: Functions, Limits and Continuity 48
Theorem 21. If limx→a
f(x) = −∞ and limx→a
g(x) = c where c is any
constant except 0, then
(i) if c > 0, limx→a f(x) · g(x) = −∞;
(ii) if c < 0, limx→a f(x) · g(x) = +∞.
The theorem holds if “x→ a” is replaced by “x→ a+” or “x→ a−.”
Chapter 2: Functions, Limits and Continuity 49
Definition 8. The line x = a is a vertical asymptote ofthe graph of the function f if at least one of the followingstatements is true.
(i) limx→a+
f(x) = +∞
(ii) limx→a+
f(x) = −∞
(iii) limx→a−
f(x) = +∞
(iv) limx→a−
f(x) = −∞
Chapter 2: Functions, Limits and Continuity 50
Example 28. Find the vertical asymptotes of the graph of the function andsketch the graph.
(a) f(x) =2
x− 4
(b) f(x) =5
x2 + 8x+ 15
(c) f(x) =2
(x− 4)2
Chapter 2: Functions, Limits and Continuity 51
Limits at Infinity
Consider the function f(x) =2x2
x2 + 1. Observe that as x increases
(decreases) without bound, the value of the function approaches 2.
Thus, we write
limx→+∞
f(x) = 2 and limx→−∞
f(x) = 2.
Chapter 2: Functions, Limits and Continuity 52
Definition 9. (Definition of the Limit of f(x) as x Increaseswithout Bound) Let f be a function defined at every numberin some interval (a,+∞). The limit of f(x) as x increaseswithout bound is L, written as
limx→+∞
f(x) = L
if for any ε > 0, however small, there exists a number N > 0such that
if x > N then |f(x)− L| < ε.
Chapter 2: Functions, Limits and Continuity 53
Definition 10. (Definition of the Limit of f(x) as xDecreases without Bound) Let f be a function definedat every number in some interval (−∞, a). The limit of f(x)as x decreases without bound is L, written as
limx→−∞
f(x) = L
if for any ε > 0, however small, there exists a number N < 0such that
if x < N then |f(x)− L| < ε.
Chapter 2: Functions, Limits and Continuity 54
Theorem 22. If r is any positive integer, then
(i) limx→+∞1
xr= 0 (ii) limx→−∞
1
xr= 0
Example 29. Find limx→+∞4x− 3
2x+ 5.
Example 30. Find limx→+∞2x2 − x+ 5
4x3 − 1.
Example 31. Find limx→+∞3x+ 4√2x2 − 5
and limx→−∞3x+ 4√2x2 − 5
.
Example 32. Find limx→+∞x2
x+ 1.
Example 33. Find limx→+∞2x− x2
3x+ 5.
Chapter 2: Functions, Limits and Continuity 55
Horizontal Asymptotes
Definition 11. (Definition of a Horizontal Asymptote) Theline y = b is a horizontal asymptote of the graph of the functionf if at least one of the following statements is true:
(i) limx→+∞ f(x) = b and for some number N , if x > N ,then f(x) 6= b;
(ii) limx→−∞ f(x) = b and for some number N , if x < N ,then f(x) 6= b.
Example 34. Find the horizontal asymptotes of the graph of the function
f(x)) =x
√x2 + 1
.
Chapter 2: Functions, Limits and Continuity 56
Chapter 2: Functions, Limits and Continuity 57
Oblique Asymptotes
Definition 12. (Definition of a function Continuous at aNumber) The graph of the function f has the line y = mx+ bas an asymptote if either of the following statements is true:
(i) limx→∞[f(x) − (mx + b)] = 0 and for some numberM > 0, f(x) 6= mx+ b whenever x > M ;
(ii) limx→−∞[f(x) − (mx + b)] = 0 and for some numberM < 0, f(x) 6= mx+ b whenever x < M .
Part (i) of the definition indicates that for any ε > 0, there exists anumber N > 0 such that
if x > N then 0 < |f(x)− (mx+ b)| < ε,
that is, we can make the function value f(x) as close to the value f(x) as
Chapter 2: Functions, Limits and Continuity 58
close to the value of mx+ b as we please by taking x sufficiently large. Asimilar statement may be made for part (ii) of the definition.
The graph of a rational functionP (x)
Q(x), where the degree of the
polynomial P (x) is one more than the degree of Q(x) and Q(x) is not a
factor of P (x), has an oblique asymptote. To show this, we let f(x) =P (x)
Q(x)and divide P (x) by Q(x) to express f(x) as the sum of a linear functionand a rational function; that is,
f(x) = mx+ b+R(x)
Q(x)
where the degree of the polynomial is less than the degree of Q(x). Then
f(x)− (mx+ b) =R(x)
Q(x).
Chapter 2: Functions, Limits and Continuity 59
When the numerator and denominator of R(x)/Q(x) is divided by thehighest power of x appearing in Q(x), there will be a constant term in thedenominator and all other terms in the denominator and every term in thenumerator will be of the form k/xr where k is a constant and r is a positiveinteger. Therefore, as x→ +∞, the limit of the numerator will be zero and
the limit of the denominator will be a constant. Thus, limx→∞R(x)
Q(x)= 0.
Hence,limx→∞
[f(x)− (mx+ b) = 0.
From this we conclude that the line y = mx+ b is an oblique asymptote ofthe graph of f .
Chapter 2: Functions, Limits and Continuity 60
Example 35. Find the asymptotes of the graph of h(x) =x2 + 3
x− 1.
Chapter 2: Functions, Limits and Continuity 61
Continuity
Definition 13. (Definition of a function Continuous at aNumber) The function f is said to be continuous at thenumber a if and only if the following three conditions aresatisfied:
(i) f(a) exists;
(ii) limx→a f(x) exists;
(iii) limx→a f(x) = f(a).
If one or more of these three conditions fails to hold at a, thefunction f is said to be discontinuous at a.
Chapter 2: Functions, Limits and Continuity 62
Continuity of the function f(x) = x+ 1 at a = 0
Chapter 2: Functions, Limits and Continuity 63
Example 36. Consider the function f(x) = x+ 1 where x 6= 0. Then f isdiscontinuous at a = 0 since f(0) does not exist.
Chapter 2: Functions, Limits and Continuity 64
Example 37. Consider the function f(x) =
{x+ 1, if x 6= 02, otherwise.
where
x 6= 0. Then f is discontinuous at a = 0 since f(0) 6= limx→a f(x).
Removable Discontinuity
Chapter 2: Functions, Limits and Continuity 65
Remark: The discontinuity described in each of the previous examples iscalled a removable discontinuity for the reason that the function can beredefined so that f(0) = 1.
In general, if f is a function discontinuous at the number a butfor which limx→a f(x) exists. Then either f(a) does not exist or elselimx→a f(x) 6= f(a). Such discontinuity is a removable discontinuitybecause f is redefined at a so that f(a) is equal to limx→a f(x), the newfunction becomes continuous at a. If the discontinuity is not removable, itis called an essential discontinuity.
Chapter 2: Functions, Limits and Continuity 66
Example 38. The function f(x) =1
x2is discontinuous at a = 0.
Infinite Discontinuity
The discontinuity of this function at the number a = 0 is essential sincelimx→a f(x) does not exist. This kind of discontinuity is called an infinitediscontinuity.
Chapter 2: Functions, Limits and Continuity 67
Example 39. The function f(x) =
{1, if x ≥ 0−1, if x < 0
is discontinuous at
a = 0.
Jump Discontinuity
The discontinuity illustrated in this example is essential sincelimx→a+ f(x) 6= limx→a− f(x) and hence, limx→a f(x) does not exist.This discontinuity is called a jump discontinuity.
Chapter 2: Functions, Limits and Continuity 68
Theorem 23. If f and g are two functions continuous at the number a,then
(i) f + g is continuous at a;
(ii) f − g is continuous at a;
(iii) f · g is continuous at a;
(iv) f/g is continuous at a provided that g(a) 6= 0.
Theorem 24. A polynomial function is continuous at every number.
Theorem 25. A rational function is continuous at every number in itsdomain.
Chapter 2: Functions, Limits and Continuity 69
Theorem 26. If n is a positive integer and
f(x) = n√x
then
1. if n is odd, f is continuous at every number.
2. if n is even, f is continuous at every positive number.
Theorem 27. (Alternative Definition of Continuity)If the function f is continuous at the number a if f is defined on someopen interval containing a and if for any ε > 0 there exists a δ > 0 suchthat
if |x− a| < δ then |f(x)− f(a)| < ε.
Chapter 2: Functions, Limits and Continuity 70
Theorem 28. (Limit of a Composite Function)If limx→a
g(x) = b and if the function f is continuous at b,
limx→a
(f ◦ g)(x) = f(b)
or equivalently,limx→a
f(g(x)) = f( limx→a
g(x))
.
Theorem 29. (Continuity of a Composite Function)If the function g is continuous at a and the function f is continuous atg(a), then the composite function f ◦ g is continuous at a.
Chapter 2: Functions, Limits and Continuity 71
Definition 14. (Definition of Continuity on an OpenInterval) A function is said to be continuous on an openinterval if and only if it is continuous at every number in theopen interval.
Definition 15. (Definition of Right-Hand Continuity) Afunction is said to be continuous from the right at the numbera if and only if the following three conditions are satisfied:
1. f(a) exists;
2. limx→a+
f(x) exists;
3. limx→a+
f(x) = f(a).
Chapter 2: Functions, Limits and Continuity 72
Definition 16. (Definition of Left-Hand Continuity) Afunction is said to be continuous from the left at the number aif and only if the following three conditions are satisfied:
1. f(a) exists;
2. limx→a−
f(x) exists;
3. limx→a−
f(x) = f(a).
Chapter 2: Functions, Limits and Continuity 73
Chapter 2: Functions, Limits and Continuity 74
Definition 17. (Definition of Continuity on a ClosedInterval) A function whose domain includes the closed interval[a, b] is said to be continuous on [a, b] if and only if it iscontinuous on the open interval (a, b), as well as continuousfrom the right at a and continuous from the left at b.
In general, a function f is right-continuous (continuous from theright) at a point x = c in its domain if limx→c+ f(x) = f(c). It is left-continuous (continuous from the left) at a point x = c in its domain iflimx→c− f(x) = f(c). Thus, a function is continuous at a left endpoint a ofits domain if it is right-continuous at a and continuous at a right endpointb of its domain if it is left-continuous at b. A function is continuous at aninterior point c of its domain if and only if it is both right-continuous andleft-continuous at c.
Chapter 2: Functions, Limits and Continuity 75
Example 40. The function f(x) =√4− x2 is continuous at every point of
its domain [−2, 2]. This includes x = −2, where f is right-continuous andx = 2, where f is left-continuous.
Example 41. The unit function U(x) =
{1, if x ≥ 00, if x < 0
, is right-
continuous at x = 0, but is neither left-continuous nor continuous there.
Chapter 2: Functions, Limits and Continuity 76
Continuity TestA function f(x) is continuous at x = a if and only if it meetsthe following three conditions:
1. f(a) exists (c lies in the domain of f)
2. limx→a f(x) exists (f has a limit as x→ a)
3. limx→a f(x) = f(a) (the limit equals the function value)
For one-sided continuity and continuity at an endpoint, the limits in (2)and (3) of the test should be replaced by the appropriate one-sided limits.
Chapter 2: Functions, Limits and Continuity 77
Example 42. Consider the function y = f(x) in the given figure, whosedomain is the closed interval [0, 4]. Discuss the continuity of f at x =0, 1, 2, 3, 4.
Chapter 2: Functions, Limits and Continuity 78
y =√4− x2
Continuous on [−2, 2]
Chapter 2: Functions, Limits and Continuity 79
y =1
xContinuous on (−∞, 0) and (0,+∞)
Chapter 2: Functions, Limits and Continuity 80
y = U(x)
Continuous on (−∞, 0) and [0,+∞)
Chapter 2: Functions, Limits and Continuity 81
y = cos x
Continuous on (−∞,+∞)
Chapter 2: Functions, Limits and Continuity 82
Definition 18. (Definition of Continuity on a Half-OpenInterval)
(i) A function whose domain includes the interval half-openinterval to the right [a, b) is continuous on [a, b) if itis continuous on the open interval (a, b) and continuousfrom the right at a.
(ii) A function whose domain includes the interval half-openinterval to the left (a, b] is continuous on (a, b] if it iscontinuous on the open interval (a, b) and continuousfrom the left at b.
Chapter 2: Functions, Limits and Continuity 83
Example 43. Determine the largest interval (or union of intervals) on whichthe following function is continuous:
f(x) =
√25− x2
x− 3
Chapter 2: Functions, Limits and Continuity 84
The Intermediate Value Theorem
Theorem 30. (The Intermediate Value Theorem (IVT).)If the function f is continuous on the closed interval [a, b], and if f(a) 6=f(b), then for any number k between f(a) and f(b) there exists a numberc between a and b such that f(c) = k.
Chapter 2: Functions, Limits and Continuity 85
In terms of geometry, the IVT states that the graph of a continuousfunction on a closed interval must intersect every horizontal line y = kbetween the lines y = f(a) and y = f(b) at least once.
The IVT also assures us that if the function is continuous on the closedinterval [a, b], then f(x) assumes every value between f(a) and f(b) as xassumes values between a and b.
Chapter 2: Functions, Limits and Continuity 86
The following is a direct consequence of the IVT.
Theorem 31. (The Intermediate-Zero Theorem.)If the function f is continuous on the closed interval [a, b], and if f(a)and f(b) have opposite signs, then there exists a number c between a andb such that f(c) = 0; that is, c is a zero of f .
Chapter 2: Functions, Limits and Continuity 87
Continuity of the Trigonometric Functions
and the Squeeze Theorem
Theorem 32. (The Squeeze Theorem.)Suppose that the functions f , g, and h are defined on some open intervalI containing a except possibly at a itself, and that f(x) ≤ g(x) ≤ h(x)for all x ∈ I for which x 6= a. Also suppose that lim
x→af(x) and lim
x→ah(x)
exist and are equal to L. Then limx→a
g(x) an is equal to L.
Chapter 2: Functions, Limits and Continuity 88
Example 44. Let the functions f , g and h be defined by
f(x) = −4(x−2)2+3, g(x) =(x− 2)(x2 − 4x+ 7)
(x− 2), h(x) = 4(x−2)2+3.
Chapter 2: Functions, Limits and Continuity 89
Example 45. Given |g(x) − 2| ≤ 3(x − 1)2 for all x. Use the SqueezeTheorem to find lim
x→2g(x).
Example 46. Use the Squeeze Theorem to prove that limx→0
∣∣∣∣∣x sin 1
x
∣∣∣∣∣ = 0.
Chapter 2: Functions, Limits and Continuity 90
Consider the function f(x) =sinx
x. This function is not defined at
x = 0 but limx→0
f(x) exists and is equal to 1.
Chapter 2: Functions, Limits and Continuity 91
Theorem 33.
limt→0
sin t
t= 1
Proof. First assume that 0 < t <π/2. The figure shows the unit circlex2 + y2 = 1 and the shaded sectorBOP , where B is the point (1, 0)and P is the point (cos t, sin t). Thearea of a circular sector of radius rand central angle of radian measure tis determined by 1
2r2t; so if S square
units is the area of sector BOP , thenS = 1
2t. Consider now the triangleBOP with area K1 square units.Hence, K1 = 1
2|AP | · |OB| =12 sin t.
Chapter 2: Functions, Limits and Continuity 92
If K2 is the area of righttriangle BOT , where T is the point(1, tan t), then K2 =
12|BT | · |OB| =
12 tan t. Observe that K1 < S < K2
from which we obtain the inequality12 sin t <
12t <
12 tan t. Multiplying
each member of this inequality by2/ sin t, which is positive because0 < t < π/2, we obtain
1 <t
sin t<
1
cos t.
By taking the reciprocal of eachmember of this inequality, obtain
1 >sin t
t> cos t.
Chapter 2: Functions, Limits and Continuity 93
From cos t <sin t
t< 1 we obtain the inequality sin t < t. By replacing
t by 12t, we obtain
sin
(1
2t
)<
1
2t.
Squaring both sides of this inequality will yield
sin2(1
2t
)<
1
4t2 or
1− cos t
2<
1
4t2.
Thus, we obtain
1− 1
2t2 < cos t.
Using this inequality and
cos t <sin t
t
Chapter 2: Functions, Limits and Continuity 94
implies
1− 1
2t2 <
sin t
t< 1 if 0 < t < π/2.
Now if −π/2 < t < 0, then 0 < −t < −π/2 so that from the aboveinequality, we obtain
1− 1
2(−t)2 <
sin(−t)(−t)
< 1 if − π/2 < t < 0.
Hence, since sin(−t) = − sin t, we have
1− 1
2t2 <
sin t
t< 1 if − π/2 < t < 0.
Therefore,
1− 1
2t2 <
sin t
t< 1 if − π/2 < t < π/2 and t 6= 0.
Chapter 2: Functions, Limits and Continuity 95
Since
limt→
(1− 1
2t2) = 1 and lim
t→1 = 1
and by applying the Squeeze Theorem we obtain the desired result
limt→0
sin t
t= 1.
Example 47. Find limx→0
sin 3x
sin 5x.
Chapter 2: Functions, Limits and Continuity 96
Theorem 34. The sine function is continuous at 0.
Theorem 35. The cosine function is continuous at 0.
Theorem 36.
limt→0
1− cos t
t= 0
Example 48. Find limx→0
1− cosx
sinx.
Example 49. Find limx→0
2 tanx
x2.
Theorem 37. The sine and cosine functions are continuous at every realnumber.
Theorem 38. The tangent, cotangent, secant, and cosecant functions arecontinuous on their domains.
Chapter 2: Functions, Limits and Continuity 97