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Benginning Calculus Lecture notes 2 - limits and continuity
1. Beginning Calculus
- Limits and Continuity -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 1 / 54
2. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Learning Outcomes
Determine the existence of limits of functions
Compute the limits of functions
Determine the continuity of functions.
Connect the idea of limits and continuity of functions.
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3. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Limits
De…nition 1
The limit of f (x), as x approaches a, equals L, denoted by
lim
x!a
f (x) = L or f (x) ! L as x ! a (1)
if the values of f (x) moves arbitrarily close to L as x moves su¢ ciently
close to a (on either side of a ) but not equal to a.
VillaRINO DoMath, FSMT-UPSI
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4. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
x!2
x2 x + 2 = 4
0 2 4
0
5
10
x
y x < 2 f (x) x > 2 f (x)
1.0 2.000000 3.0 8.000000
1.5 2.750000 2.5 5.750000
1.9 3.710000 2.1 4.310000
1.99 3.970100 2.01 4.030100
1.995 3.985025 2.005 4.015025
1.999 3.997001 2.001 4.003001
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5. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
Estimate the value of lim
t!0
p
t2 + 9 3
t2
.
f
0
B
B
B
B
B
B
B
B
B
B
B
B
@
t
0.1
0.001
0.0001
0.00001
0.00001
0.0001
0.001
0.1
1
C
C
C
C
C
C
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
t2
p
t2 + 9 3
0.166 62
0.166 67
0.166 67
0.166 67
0.166 67
0.166 67
0.166 67
0.166 62
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
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6. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example - continue
-4 -2 0 2 4
0.12
0.13
0.14
0.15
0.16
lim
t!0
p
t2 + 9 3
t2
=
1
6
VillaRINO DoMath, FSMT-UPSI
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7. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
f (x) = x + 1.
-1 1 2 3 4 5
-1
1
2
3
4
5
x
y
lim
x!2
f (x) = 3
VillaRINO DoMath, FSMT-UPSI
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8. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
g (x) =
x + 1 if x 2
(x 2)2
+ 3 if x > 2
-1 1 2 3 4 5
-1
1
2
3
4
5
x
y
lim
x!2
g (x) = 3
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9. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
h (x) =
x + 1 if x < 2
(x 2)2
+ 3 if x > 2
-1 1 2 3 4 5
-1
1
2
3
4
5
x
y
lim
x!2
h (x) = 3, eventhough h is not de…ned at x = 2.
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10. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
One-Sided Limits
Left-hand limit of f
lim
x!a
f (x) = L (2)
Right-hand limit of f
lim
x!a+
f (x) = L (3)
lim
x!a
f (x) = L , f lim
x!a
f (x) = lim
x!a+
f (x) = L. (4)
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11. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
f (x) =
x + 1 if x 2
(x 2)2
+ 1 if x > 2
-1 1 2 3 4 5
-1
1
2
3
4
5
x
y
lim
x!2
f (x) = 3 and lim
x!2+
f (x) = 1
lim
x!2
f (x) does not exist (DNE), eventhough f is de…ned at x = 2.
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12. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
-1 1 2 3 4 5
-1
1
2
3
4
5
x
y
Find:
f (2) and f (4)
lim
x!2
f (x) , lim
x!2+
f (x) , lim
x!2
f (x)
lim
x!4
f (x) , lim
x!4+
f (x) lim
x!4
f (x)
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13. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Properties of Limits
Suppose that lim
x!a
f (x) and lim
x!a
g (x) exists. Then,
1. lim
x!a
(cf (x)) = c lim
x!a
f (x) , for any constant c
2. lim
x!a
[f (x) g (x)] = lim
x!a
f (x) lim
x!a
g (x)
3. lim
x!a
[f (x) g (x)] =
h
lim
x!a
f (x)
i h
lim
x!a
g (x)
i
4. lim
x!a
f (x)
g (x)
=
lim
x!a
f (x)
lim
x!a
g (x)
provided that lim
x!a
g (x) 6= 0
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14. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Properties of Limits - continue
5. lim
x!a
x = a
6. lim
x!a
c = c, for any constant c.
7. lim
x!a
[f (x)]n
=
h
lim
x!a
f (x)
in
where n 2 Z+.
8. lim
x!a
n
p
x = n
p
a where n 2 Z+ (If n is even, we assume that a > 0 ).
9. lim
x!a
n
p
f (x) = n
q
lim
x!a
f (x) where n 2 Z+. (If n is even, we assume
that lim
x!a
f (x) > 0 ).
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15. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Direct Substitution Property
If f is a polynomial or a rational function and a is in the domain of f ,
then
lim
x!a
f (x) = f (a) (5)
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16. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
x!5
2x2
3x + 4 = lim
x!5
2x2
lim
x!5
3x + lim
x!5
4
= 2 lim
x!5
x2
3 lim
x!5
x + lim
x!5
4
= 2 52
3 (5) + 4
= 39
VillaRINO DoMath, FSMT-UPSI
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17. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
x! 2
x3 + 2x2 1
5 3x
=
lim
x! 2
x3 + 2x2 1
lim
x! 2
(5 3x)
=
lim
x! 2
x3 + 2 lim
x! 2
x2 lim
x! 2
1
lim
x! 2
5 3 lim
x! 2
x
=
( 2)3
+ 2 ( 2)2
1
5 3 ( 2)
=
1
11
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18. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
De…nition 2
If f (x) = g (x) when x 6= a, then lim
x!a
f (x) = lim
x!a
g (x) , provided that
the limits exist.
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19. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
x!1
x2 1
x 1
. For x 6= 1,
x2 1
x 1
=
(x 1) (x + 1)
x 1
= x + 1
lim
x!1
x2 1
x 1
= lim
x!1
x + 1 = 2
-1 1 2 3 4 5
-1
1
2
3
4
5
x
y
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20. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
h!0
(3 + h)2
9
h
. For h 6= 0,
(3 + h)2
9
h
=
9 + 6h + h2 9
h
= 6 + h
lim
h!0
(3 + h)2
9
h
= lim
h!0
6 + h = 6
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21. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
x!2
jx 2j
x 2
.
For x 2 > 0, jx 2j = x 2.
lim
x!2
jx 2j
x 2
= lim
x!2
x 2
x 2
= lim
x!2
1 = 1
For x 2 < 0, jx 2j = (x 2) = 2 x.
lim
x!2
jx 2j
x 2
= lim
x!2
(x 2)
x 2
= lim
x!2
1 = 1
lim
x!2
jx 2j
x 2
DNE
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22. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Remark 1
lim
θ!0
cos θ 1
θ
= 0
Rewrite:
1 cos θ
θ
to make the numerator stays positive.
θ
1
O
A
BC
BC = 1 cos θ, arclength AB = θ.
1 cos θ
θ
! 0 as θ ! 0
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23. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Remark 2
lim
θ!0
sin θ
θ
= 1
θ
1
O
A
BC
AC = sin θ, arclength AB = θ
sin θ
θ
! 1 as θ ! 0.
Principle: Short pieces of curves are nearly straight.
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24. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
θ!0
tan θ
θ
tan θ
θ
=
sin θ
cos θ
θ
=
sin θ
θ cos θ
=
sin θ
θ
1
cos θ
lim
θ!0
tan θ
θ
= lim
θ!0
sin θ
θ
lim
θ!0
1
cos θ
= 1 1 = 1
VillaRINO DoMath, FSMT-UPSI
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25. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
θ!0
sin 2θ
tan θ
sin 2θ
tan θ
=
sin 2θ
θ
tan θ
θ
=
2 sin 2θ
2θ
tan θ
θ
lim
θ!0
sin 2θ
tan θ
= lim
θ!0
2 sin 2θ
2θ
tan θ
θ
=
lim
θ!0
2 sin 2θ
2θ
lim
θ!0
tan θ
θ
=
2
1
= 2
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26. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
In…nite Limits
De…nition 3
Let f de…ned on both sides of a, except possibly at a itself. Then
lim
x!a
f (x) = ∞ or lim
x!a
f (x) = ∞ (6)
means that the values of f (x) can be made arbitrarily large (as large as
possible) by taking x su¢ ciently close to a, but not equal to a. x = a is
the vertical asymptote.
y
x
y = f(x)
x = a
a
y
x
y = f(x)
x = a
a
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27. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
x!3+
2x
x 3
= +∞ and lim
x!3
2x
x 3
= ∞
-5 5 10
-5
5
10
x
y
x = 3
The vertical asymptote is at x = 3.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 27 / 54
28. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
f (x) = tan x =
sin x
cos x
The vertical asymptote can be obtained by setting cos x = 0, that is,
x =
π
2
x = (2n + 1)
π
2
, n 2 Z
x
y
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29. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Limits at In…nity
De…nition 4 (Limits at In…nity)
(a) Let f be a function de…ned on some interval (a, ∞) . Then
lim
x!∞
f (x) = L (7)
means that the values of f (x) can be made arbitrarily close to L by
taking x su¢ ciently large.
(b) Let f be a function de…ned on some interval ( ∞, a) . Then
lim
x! ∞
f (x) = L (8)
means that the values of f (x) can be made arbitrarily close to L by
taking x su¢ ciently large negative.
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30. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Horizontal Asymptotes
The line y = L is called a horizontal asymptote of the curve y = f (x) if
either
lim
x!∞
f (x) = L or lim
x! ∞
f (x) = L (9)
VillaRINO DoMath, FSMT-UPSI
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31. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
f (x) =
x2 1
x2 + 1
lim
x!∞
f (x) = 1 = lim
x! ∞
f (x)
-10 -5 5 10
-1
1
2
x
y
No vertical asymtote.
The horizontal asymptote is y = 1.
VillaRINO DoMath, FSMT-UPSI
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32. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
f (x) =
1
x
.
lim
x!0
1
x
= ∞, lim
x!0+
1
x
= +∞
lim
x!∞
1
x
= 0 = lim
x! ∞
1
x
Vertical asymtote at x = 0
The horizontal asymptote at y = 0.
-4 -2 2 4
-4
-2
2
4
x
y
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33. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example - Finding the Asymptotes
f (x) =
3x2 x 2
5x2 + 4x + 1
lim
x!∞
3x2 x 2
5x2 + 4x + 1
= lim
x!∞
3x2
x2
x
x2
2
x2
5x2
x2
+
4x
x2
+
1
x2
= lim
x!∞
3
1
x
2
x2
5 +
4
x
+
1
x2
=
lim
x!∞
3
1
x
2
x2
lim
x!∞
5 +
4
x
+
1
x2
=
lim
x!∞
3 lim
x!∞
1
x
lim
x!∞
2
x2
lim
x!∞
5 + lim
x!∞
4
x
+ lim
x!∞
1
x2
=
3 0 0
5 + 0 + 0
=
3
5
The horizontal asymptote is y =
3
5
.
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34. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example - Finding the Asymptotes
f (x) =
p
2x2 + 1
3x 5
.
lim
x!∞
p
2x2 + 1
3x 5
= lim
x!∞
p
2x2 + 1
p
x2
3x 5
x
,
p
x2 = x for x > 0
= lim
x!∞
r
2x2
x2
+
1
x2
3x
x
5
x
= lim
x!∞
r
2 +
1
x2
3
5
x
=
lim
x!∞
r
2 +
1
x2
lim
x!∞
3
5
x
=
r
lim
x!∞
2 + lim
x!∞
1
x2
lim
x!∞
3 lim
x!∞
5
x
=
p
2
3
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35. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example - Finding the Asymptotes - continue
lim
x! ∞
p
2x2 + 1
3x 5
= lim
x! ∞
r
2 +
1
x2
3
5
x
,
p
x2 = x for x < 0
=
lim
x!∞
r
2 +
1
x2
lim
x! ∞
3
5
x
=
p
2
3
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 35 / 54
36. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example - Finding the Asymptotes - continue
-4 -2 2 4
-4
-2
2
4
x
y
The horizontal asymptotes are: y =
p
2
3
.
The vertical asymptote is when 3x 5 = 0, that is, x =
5
3
.
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37. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example - Finding the Asymptotes
f (x) =
p
x2 + 1 x
lim
x!∞
p
x2 + 1 x = lim
x!∞
p
x2 + 1 x
p
x2 + 1 + x
p
x2 + 1 + x
= lim
x!∞
x2 + 1 x2
p
x2 + 1 + x
= lim
x!∞
1
p
x2 + 1 + x
= lim
x!∞
1
xp
x2 + 1 + x
p
x2
= lim
x!∞
1
xr
x2
x2
+
1
x2
+ 1
= lim
x!∞
1
xr
1 +
1
x2
+ 1
=
0
p
1 + 0 + 1
= 0
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(D1) Limits and Continuity 37 / 54
38. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example - Finding the Asymptotes - continue
-4 -2 0 2 4
5
10
x
y
The horizontal asymptote is y = 0.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 38 / 54
39. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
x!∞
x3 = ∞ and lim
x! ∞
x3 = ∞.
-4 -2 2 4
-100
-50
50
100
x
y
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40. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
x!∞
x2 x . Note that the properties of limits cannot be applied to
in…nite limits since ∞ is not a number. So,
lim
x!∞
x2
x = lim
x!∞
x (x 1) = ∞
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41. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
lim
x!∞
x2 + x
3 x
.
lim
x!∞
x2 + x
3 x
= lim
x!∞
x2
x
+
x
x
3
x
x
x
= lim
x!∞
x + 1
3
x
1
=
∞
1
= ∞
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42. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Continuous Functions at a Point
De…nition 5
A function f is continuous at a if
lim
x!a
f (x) = f (a) (10)
y
x
y = f(x)
a
f(a)
f (a) is de…ned (a is in the domain of f )
lim
x!a
f (x) exists.
lim
x!a
f (x) = f (a)
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43. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
y
x1 3 50 2 4 6
Discontinuities at 1, 3, and 5.
at a = 1, f is unde…ned
at a = 3, f is de…ned but lim
x!3
f (x) DNE;
at a = 5, f is de…ned and lim
x!5
f (x) exists, but lim
x!5
f (x) 6= f (5) .
VillaRINO DoMath, FSMT-UPSI
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44. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
f (x) =
x2 x 2
x 2
is discontinuous at 2 because f (2) is unde…ned.
-1 1 2 3 4 5
-1
1
2
3
4
5
x
y
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45. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
g (x) =
( 1
x2
if x 6= 0
1 if x = 0
is de…ned at 0 but lim
x!0
g (x) = lim
x!0
1
x2
does not exist. This discontinuity is called in…nite discontinuity.
-4 -2 2 4
-1
1
2
3
4
5
x
y
VillaRINO DoMath, FSMT-UPSI
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46. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
h (x) =
8
<
:
x2 x 2
x 2
if x 6= 2
1 if x = 2
is de…ned at 2 and lim
x!2
h (x) = 3,
but lim
x!2
h (x) 6= h (2) . This discontinuity is called removable
discontinuity.
-1 1 2 3 4 5
-1
1
2
3
4
5
x
y
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 46 / 54
47. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
k (x) = bxc has discontinuities at all of the integers because lim
x!n
k (x)
does not exist if n is an integer. These discontinuities are called jump
discontinuities.
-1 1 2 3 4 5
-1
1
2
3
4
5
x
y
VillaRINO DoMath, FSMT-UPSI
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48. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Theorem 6
If f and g are continuous at x = a and c is a constant, then the
following functions are also continuous at a.
(a) f g
(b) cf
(c) fg
(d)
f
g
if g (a) 6= 0
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 48 / 54
49. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Theorem 7
The following functions are continuous at every number in their domains.
(a) Polynomial functions.
(b) Rational functions.
(c) Power and root functions
(d) Trigonometric Functions
VillaRINO DoMath, FSMT-UPSI
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50. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
f (x) = x100 2x37 + 75 is a polynomial function. So it is
continuous everywhere: ( ∞, ∞)
g (x) =
x2 + 2x + 17
x2 1
is a rational function, and continuous on its
domain fx j x 6= 1g = ( ∞, 1) [ ( 1, 1) [ (1, ∞) .
VillaRINO DoMath, FSMT-UPSI
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51. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
h (x) =
p
x +
x + 1
x 1
x + 1
x2 + 1
Let h1 (x) =
p
x; h2 (x) =
x + 1
x 1
; and h3 (x) =
x + 1
x2 + 1
.
h1 (x) is a root function and continuous on [0, ∞).
h2 (x) is a rational function and continuous on ( ∞, 1) [ (1, ∞) ,
and
h3 (x) is also a rational function and continuous everywhere on R.
So, h (x) is continuous on [0, 1) [ (1, ∞) .
VillaRINO DoMath, FSMT-UPSI
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52. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
f (x) =
sin x
2 + cos x
Let f1 (x) = sin x, and let f2 (x) = 2 + cos x.
f1 (x) and f2 (x) are trigonometric functions. So, they are
continuous. Note that cos x 1. So, f2 (x) = 2 cos x is always
positive.
Hence, f (x) =
f1 (x)
f2 (x)
=
sin x
2 + cos x
is continuous everywhere on R.
VillaRINO DoMath, FSMT-UPSI
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53. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Theorem 8
If g is continuous at a and f is continuous at g (a) , then
(f g) (x) = f (g (x)) is continuous at a.
VillaRINO DoMath, FSMT-UPSI
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54. The Limit of a Function Limits of Trigonometric Functions In…nite Limits and Limits at In…nity Continuity
Example
f (x) = sin x2
Let F (x) = sin x, and let G (x) = x2.
F and G are continuous on R.
So, f (x) = F (G (x)) = sin x2 is continuous on R.
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