1. Section 1.4
Calculating Limits
V63.0121.002.2010Su, Calculus I
New York University
May 18, 2010
Announcements
WebAssign Class Key: nyu 0127 7953
Office Hours: MR 5:00–5:45, TW 7:50–8:30, CIWW 102 (here)
Quiz 1 Thursday on 1.1–1.4
. . . . . .
2. Announcements
WebAssign Class Key: nyu
0127 7953
Office Hours: MR
5:00–5:45, TW 7:50–8:30,
CIWW 102 (here)
Quiz 1 Thursday on
1.1–1.4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 2 / 37
3. Objectives
Know basic limits like
lim x = a and lim c = c.
x→a x→a
Use the limit laws to
compute elementary limits.
Use algebra to simplify
limits.
Understand and state the
Squeeze Theorem.
Use the Squeeze Theorem
to demonstrate a limit.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 3 / 37
4. Outline
Basic Limits
Limit Laws
The direct substitution property
Limits with Algebra
Two more limit theorems
Two important trigonometric limits
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 4 / 37
5. Really basic limits
Fact
Let c be a constant and a a real number.
(i) lim x = a
x→a
(ii) lim c = c
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 5 / 37
6. Really basic limits
Fact
Let c be a constant and a a real number.
(i) lim x = a
x→a
(ii) lim c = c
x→a
Proof.
The first is tautological, the second is trivial.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 5 / 37
9. ET game for f(x) = x
y
.
. .
a
. . x
.
a
.
. . . . . .
10. ET game for f(x) = x
y
.
. .
a
. . x
.
a
.
. . . . . .
11. ET game for f(x) = x
y
.
. .
a
. . x
.
a
.
. . . . . .
12. ET game for f(x) = x
y
.
. .
a
. . x
.
a
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 6 / 37
13. ET game for f(x) = x
y
.
. .
a
. . x
.
a
.
Setting error equal to tolerance works!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 6 / 37
17. ET game for f(x) = c
y
.
.
c
.
. . x
.
a
.
. . . . . .
18. ET game for f(x) = c
y
.
.
c
.
. . x
.
a
.
. . . . . .
19. ET game for f(x) = c
y
.
.
c
.
. . x
.
a
.
. . . . . .
20. ET game for f(x) = c
y
.
.
c
.
. . x
.
a
.
any tolerance works!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 7 / 37
21. Really basic limits
Fact
Let c be a constant and a a real number.
(i) lim x = a
x→a
(ii) lim c = c
x→a
Proof.
The first is tautological, the second is trivial.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 8 / 37
22. Outline
Basic Limits
Limit Laws
The direct substitution property
Limits with Algebra
Two more limit theorems
Two important trigonometric limits
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 9 / 37
23. Limits and arithmetic
Fact
Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then
x→a x→a
1. lim [f(x) + g(x)] = L + M
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 10 / 37
24. Limits and arithmetic
Fact
Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then
x→a x→a
1. lim [f(x) + g(x)] = L + M (errors add)
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 10 / 37
25. Limits and arithmetic
Fact
Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then
x→a x→a
1. lim [f(x) + g(x)] = L + M (errors add)
x→a
2. lim [f(x) − g(x)] = L − M
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 11 / 37
26. Limits and arithmetic
Fact
Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then
x→a x→a
1. lim [f(x) + g(x)] = L + M (errors add)
x→a
2. lim [f(x) − g(x)] = L − M
x→a
3. lim [cf(x)] = cL
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 11 / 37
27. Limits and arithmetic
Fact
Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then
x→a x→a
1. lim [f(x) + g(x)] = L + M (errors add)
x→a
2. lim [f(x) − g(x)] = L − M
x→a
3. lim [cf(x)] = cL (error scales)
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 11 / 37
28. Justification of the scaling law
errors scale: If f(x) is e away from L, then
(c · f(x) − c · L) = c · (f(x) − L) = c · e
That is, (c · f)(x) is c · e away from cL,
So if Player 2 gives us an error of 1 (for instance), Player 1 can
use the fact that lim f(x) = L to find a tolerance for f and g
x→a
corresponding to the error 1/c.
Player 1 wins the round.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 12 / 37
29. Limits and arithmetic
Fact
Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then
x→a x→a
1. lim [f(x) + g(x)] = L + M (errors add)
x→a
2. lim [f(x) − g(x)] = L − M (combination of adding and scaling)
x→a
3. lim [cf(x)] = cL (error scales)
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 13 / 37
30. Limits and arithmetic
Fact
Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then
x→a x→a
1. lim [f(x) + g(x)] = L + M (errors add)
x→a
2. lim [f(x) − g(x)] = L − M (combination of adding and scaling)
x→a
3. lim [cf(x)] = cL (error scales)
x→a
4. lim [f(x)g(x)] = L · M
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 14 / 37
31. Limits and arithmetic
Fact
Suppose lim f(x) = L and lim g(x) = M and c is a constant. Then
x→a x→a
1. lim [f(x) + g(x)] = L + M (errors add)
x→a
2. lim [f(x) − g(x)] = L − M (combination of adding and scaling)
x→a
3. lim [cf(x)] = cL (error scales)
x→a
4. lim [f(x)g(x)] = L · M (more complicated, but doable)
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 14 / 37
32. Limits and arithmetic II
Fact (Continued)
f(x) L
5. lim = , if M ̸= 0.
x→a g(x) M
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 15 / 37
33. Caution!
The quotient rule for limits says that if lim g(x) ̸= 0, then
x→a
f(x) limx→a f(x)
lim =
x→a g(x) limx→a g(x)
It does NOT say that if lim g(x) = 0, then
x→a
f(x)
lim does not exist
x→a g(x)
In fact, it can.
more about this later
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 16 / 37
34. Limits and arithmetic II
Fact (Continued)
f(x) L
5. lim = , if M ̸= 0.
g(x)
x→a M
[ ]n
n
6. lim [f(x)] = lim f(x)
x→a x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 17 / 37
35. Limits and arithmetic II
Fact (Continued)
f(x) L
5. lim = , if M ̸= 0.
g(x)
x→a M
[ ]n
n
6. lim [f(x)] = lim f(x) (follows from 4 repeatedly)
x→a x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 17 / 37
36. Limits and arithmetic II
Fact (Continued)
f(x) L
5. lim = , if M ̸= 0.
g(x)
x→a M
[ ]n
n
6. lim [f(x)] = lim f(x) (follows from 4 repeatedly)
x→a x→a
n n
7. lim x = a
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 17 / 37
37. Limits and arithmetic II
Fact (Continued)
f(x) L
5. lim = , if M ̸= 0.
g(x)
x→a M
[ ]n
n
6. lim [f(x)] = lim f(x) (follows from 4 repeatedly)
x→a x→a
n n
7. lim x = a
x→a
√ √
8. lim n x = n a
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 17 / 37
38. Limits and arithmetic II
Fact (Continued)
f(x) L
5. lim = , if M ̸= 0.
g(x)
x→a M
[ ]n
n
6. lim [f(x)] = lim f(x) (follows from 4 repeatedly)
x→a x→a
n n
7. lim x = a (follows from 6)
x→a
√ √
8. lim n x = n a
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 17 / 37
39. Limits and arithmetic II
Fact (Continued)
f(x) L
5. lim = , if M ̸= 0.
g(x)
x→a M
[ ]n
n
6. lim [f(x)] = lim f(x) (follows from 4 repeatedly)
x→a x→a
n n
7. lim x = a (follows from 6)
x→a
√ √
8. lim n x = n a
x→a
√ √
9. lim n f(x) = n lim f(x) (If n is even, we must additionally assume
x→a x→a
that lim f(x) > 0)
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 17 / 37
40. Applying the limit laws
Example
( )
Find lim x2 + 2x + 4 .
x→3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 18 / 37
41. Applying the limit laws
Example
( )
Find lim x2 + 2x + 4 .
x→3
Solution
By applying the limit laws repeatedly:
( )
lim x2 + 2x + 4
x→3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 18 / 37
46. Your turn
Example
x2 + 2x + 4
Find lim
x→3 x3 + 11
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 19 / 37
47. Your turn
Example
x2 + 2x + 4
Find lim
x→3 x3 + 11
Solution
19 1
The answer is = .
38 2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 19 / 37
48. Direct Substitution Property
Theorem (The Direct Substitution Property)
If f is a polynomial or a rational function and a is in the domain of f, then
lim f(x) = f(a)
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 20 / 37
49. Outline
Basic Limits
Limit Laws
The direct substitution property
Limits with Algebra
Two more limit theorems
Two important trigonometric limits
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 21 / 37
50. Limits do not see the point! (in a good way)
Theorem
If f(x) = g(x) when x ̸= a, and lim g(x) = L, then lim f(x) = L.
x→a x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 22 / 37
51. Limits do not see the point! (in a good way)
Theorem
If f(x) = g(x) when x ̸= a, and lim g(x) = L, then lim f(x) = L.
x→a x→a
Example
x2 + 2x + 1
Find lim , if it exists.
x→−1 x+1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 22 / 37
52. Limits do not see the point! (in a good way)
Theorem
If f(x) = g(x) when x ̸= a, and lim g(x) = L, then lim f(x) = L.
x→a x→a
Example
x2 + 2x + 1
Find lim , if it exists.
x→−1 x+1
Solution
x2 + 2x + 1
Since = x + 1 whenever x ̸= −1, and since
x+1
x2 + 2x + 1
lim x + 1 = 0, we have lim = 0.
x→−1 x→−1 x+1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 22 / 37
53. x2 + 2x + 1
ET game for f(x) =
x+1
y
.
. . x
.
−
. 1
Even if f(−1) were something else, it would not effect the limit.
. . . . . .
54. x2 + 2x + 1
ET game for f(x) =
x+1
y
.
. . x
.
−
. 1
Even if f(−1) were something else, it would not effect the limit.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 23 / 37
55. Limit of a function defined piecewise at a boundary
point
Example
Let
{
x2 x ≥ 0 .
f(x) =
−x x < 0
Does lim f(x) exist?
x→0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 24 / 37
56. Limit of a function defined piecewise at a boundary
point
Example
Let
{
x2 x ≥ 0 .
f(x) =
−x x < 0
Does lim f(x) exist?
x→0
Solution
We have
MTP DSP
lim+ f(x) = lim+ x2 = 02 = 0
x→0 x→0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 24 / 37
57. Limit of a function defined piecewise at a boundary
point
Example
Let
{
x2 x ≥ 0 .
f(x) =
−x x < 0
Does lim f(x) exist?
x→0
Solution
We have
MTP DSP
lim+ f(x) = lim+ x2 = 02 = 0
x→0 x→0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 24 / 37
58. Limit of a function defined piecewise at a boundary
point
Example
Let
{
x2 x ≥ 0 .
f(x) =
−x x < 0
Does lim f(x) exist?
x→0
Solution
We have
MTP DSP
lim+ f(x) = lim+ x2 = 02 = 0
x→0 x→0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 24 / 37
59. Limit of a function defined piecewise at a boundary
point
Example
Let
{
x2 x ≥ 0 .
f(x) =
−x x < 0
Does lim f(x) exist?
x→0
Solution
We have
MTP DSP
lim+ f(x) = lim+ x2 = 02 = 0
x→0 x→0
Likewise:
lim f(x) = lim −x = −0 = 0
x→0− x→0− . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 24 / 37
60. Limit of a function defined piecewise at a boundary
point
Example
Let
{
x2 x ≥ 0 .
f(x) =
−x x < 0
Does lim f(x) exist?
x→0
Solution
We have
MTP DSP
lim+ f(x) = lim+ x2 = 02 = 0
x→0 x→0
Likewise:
lim f(x) = lim −x = −0 = 0
x→0− x→0− . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 24 / 37
61. Limit of a function defined piecewise at a boundary
point
Example
Let
{
x2 x ≥ 0 .
f(x) =
−x x < 0
Does lim f(x) exist?
x→0
Solution
We have
MTP DSP
lim+ f(x) = lim+ x2 = 02 = 0
x→0 x→0
Likewise:
lim f(x) = lim −x = −0 = 0
x→0− x→0− . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 24 / 37
62. Finding limits by algebraic manipulations
Example
√
x−2
Find lim .
x→4 x − 4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 25 / 37
63. Finding limits by algebraic manipulations
Example
√
x−2
Find lim .
x→4 x − 4
Solution
√ 2 √ √
Write the denominator as x − 4 = x − 4 = ( x − 2)( x + 2).
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 25 / 37
64. Finding limits by algebraic manipulations
Example
√
x−2
Find lim .
x→4 x − 4
Solution
√ 2 √ √
Write the denominator as x − 4 = x − 4 = ( x − 2)( x + 2). So
√ √
x−2 x−2
lim = lim √ √
x→4 x − 4 x→4 ( x − 2)( x + 2)
1 1
= lim √ =
x→4 x+2 4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 25 / 37
65. Your turn
Example
Let
{
1 − x2 x≥1
f(x) =
2x x<1
Find lim f(x) if it exists.
x→1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 26 / 37
66. Your turn
Example
Let
{
1 − x2 x≥1
f(x) =
2x x<1
Find lim f(x) if it exists.
x→1
Solution
We have
( )
DSP
lim+ f(x) = lim+ 1 − x2 = 0
x→1 x→1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 26 / 37
67. Your turn
Example
Let
{
1 − x2 x≥1
f(x) =
2x x<1
. .
Find lim f(x) if it exists. 1
.
x→1
Solution
We have
( )
DSP
lim+ f(x) = lim+ 1 − x2 = 0
x→1 x→1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 26 / 37
68. Your turn
Example
Let
{
1 − x2 x≥1
f(x) =
2x x<1
. .
Find lim f(x) if it exists. 1
.
x→1
Solution
We have
( )
DSP
lim+ f(x) = lim+ 1 − x2 = 0
x→1 x→1
DSP
lim f(x) = lim (2x) = 2
− −
x→1 x→1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 26 / 37
69. Your turn
Example
.
Let
{
1 − x2 x≥1
f(x) =
2x x<1
. .
Find lim f(x) if it exists. 1
.
x→1
Solution
We have
( )
DSP
lim+ f(x) = lim+ 1 − x2 = 0
x→1 x→1
DSP
lim f(x) = lim (2x) = 2
− −
x→1 x→1
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 26 / 37
70. Your turn
Example
.
Let
{
1 − x2 x≥1
f(x) =
2x x<1
. .
Find lim f(x) if it exists. 1
.
x→1
Solution
We have
( )
DSP
lim+ f(x) = lim+ 1 − x2 = 0
x→1 x→1
DSP
lim f(x) = lim (2x) = 2
− −
x→1 x→1
The left- and right-hand limits disagree, so the limit does not exist. . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 26 / 37
71. A message from the Mathematical Grammar Police
Please do not say “ lim f(x) = DNE.” Does not compute.
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 27 / 37
72. A message from the Mathematical Grammar Police
Please do not say “ lim f(x) = DNE.” Does not compute.
x→a
Too many verbs
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 27 / 37
73. A message from the Mathematical Grammar Police
Please do not say “ lim f(x) = DNE.” Does not compute.
x→a
Too many verbs
Leads to FALSE limit laws like “If lim f(x) DNE and lim g(x) DNE,
x→a x→a
then lim (f(x) + g(x)) DNE.”
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 27 / 37
74. Two More Important Limit Theorems
Theorem
If f(x) ≤ g(x) when x is near a (except possibly at a), then
lim f(x) ≤ lim g(x)
x→a x→a
(as usual, provided these limits exist).
Theorem (The Squeeze/Sandwich/Pinching Theorem)
If f(x) ≤ g(x) ≤ h(x) when x is near a (as usual, except possibly at a),
and
lim f(x) = lim h(x) = L,
x→a x→a
then
lim g(x) = L.
x→a
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 28 / 37
75. Using the Squeeze Theorem
We can use the Squeeze Theorem to replace complicated expressions
with simple ones when taking the limit.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 29 / 37
76. Using the Squeeze Theorem
We can use the Squeeze Theorem to replace complicated expressions
with simple ones when taking the limit.
Example
(π )
Show that lim x2 sin = 0.
x→0 x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 29 / 37
77. Using the Squeeze Theorem
We can use the Squeeze Theorem to replace complicated expressions
with simple ones when taking the limit.
Example
(π )
Show that lim x2 sin = 0.
x→0 x
Solution
We have for all x,
(π ) (π )
−1 ≤ sin ≤ 1 =⇒ −x2 ≤ x2 sin ≤ x2
x x
The left and right sides go to zero as x → 0.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 29 / 37
78. Illustration of the Squeeze Theorem
y
. . (x) = x2
h
. x
.
. . . . . .
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79. Illustration of the Squeeze Theorem
y
. . (x) = x2
h
. x
.
.(x) = −x2
f
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 30 / 37
80. Illustration of the Squeeze Theorem
y
. . (x) = x2
h
(π )
. (x) = x2 sin
g
x
. x
.
.(x) = −x2
f
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 30 / 37
81. Outline
Basic Limits
Limit Laws
The direct substitution property
Limits with Algebra
Two more limit theorems
Two important trigonometric limits
. . . . . .
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82. Two important trigonometric limits
Theorem
The following two limits hold:
sin θ
lim =1
θ→0 θ
cos θ − 1
lim =0
θ→0 θ
. . . . . .
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83. Proof of the Sine Limit
Proof.
Notice
θ
.
θ
.
θ
.
1
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 33 / 37
84. Proof of the Sine Limit
Proof.
Notice
sin θ ≤ θ
. in θ .
s θ
.
θ
.
c
. os θ 1
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 33 / 37
85. Proof of the Sine Limit
Proof.
Notice
sin θ ≤ θ tan θ
. in θ . .an θ
s θ t
.
θ
.
c
. os θ 1
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 33 / 37
86. Proof of the Sine Limit
Proof.
Notice
θ
sin θ ≤ θ ≤ 2 tan ≤ tan θ
2
. in θ . .an θ
s θ t
.
θ
.
c
. os θ 1
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 33 / 37
87. Proof of the Sine Limit
Proof.
Notice
θ
sin θ ≤ θ ≤ 2 tan ≤ tan θ
2
Divide by sin θ:
θ 1
1≤ ≤
sin θ cos θ
. in θ . .an θ
s θ t
.
θ
.
c
. os θ 1
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 33 / 37
88. Proof of the Sine Limit
Proof.
Notice
θ
sin θ ≤ θ ≤ 2 tan ≤ tan θ
2
Divide by sin θ:
θ 1
1≤ ≤
sin θ cos θ
. in θ . .an θ
s θ t
.
θ Take reciprocals:
.
c
. os θ 1
. sin θ
1≥ ≥ cos θ
θ
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 33 / 37
89. Proof of the Sine Limit
Proof.
Notice
θ
sin θ ≤ θ ≤ 2 tan ≤ tan θ
2
Divide by sin θ:
θ 1
1≤ ≤
sin θ cos θ
. in θ . .an θ
s θ t
.
θ Take reciprocals:
.
c
. os θ 1
. sin θ
1≥ ≥ cos θ
θ
As θ → 0, the left and right sides tend to 1. So, then, must the middle
expression.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 33 / 37
90. Proof of the Cosine Limit
Proof.
1 − cos θ 1 − cos θ 1 + cos θ 1 − cos2 θ
= · =
θ θ 1 + cos θ θ(1 + cos θ)
sin2 θ sin θ sin θ
= = ·
θ(1 + cos θ) θ 1 + cos θ
So
( ) ( )
1 − cos θ sin θ sin θ
lim = lim · lim
θ→0 θ θ→0 θ θ→0 1 + cos θ
= 1 · 0 = 0.
. . . . . .
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91. Try these
Example
tan θ
1. lim
θ→0 θ
sin 2θ
2. lim
θ→0 θ
. . . . . .
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92. Try these
Example
tan θ
1. lim
θ→0 θ
sin 2θ
2. lim
θ→0 θ
Answer
1. 1
2. 2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 35 / 37
93. Solutions
1. Use the basic trigonometric limit and the definition of tangent.
tan θ sin θ sin θ 1 1
lim = lim = lim · lim = 1 · = 1.
θ→0 θ θ→0 θ cos θ θ→0 θ θ→0 cos θ 1
. . . . . .
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94. Solutions
1. Use the basic trigonometric limit and the definition of tangent.
tan θ sin θ sin θ 1 1
lim = lim = lim · lim = 1 · = 1.
θ→0 θ θ→0 θ cos θ θ→0 θ θ→0 cos θ 1
2. Change the variable:
sin 2θ sin 2θ sin 2θ
lim = lim = 2 · lim =2·1=2
θ→0 θ 2θ→0 2θ · 1 2θ→0 2θ
2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 36 / 37
95. Solutions
1. Use the basic trigonometric limit and the definition of tangent.
tan θ sin θ sin θ 1 1
lim = lim = lim · lim = 1 · = 1.
θ→0 θ θ→0 θ cos θ θ→0 θ θ→0 cos θ 1
2. Change the variable:
sin 2θ sin 2θ sin 2θ
lim = lim = 2 · lim =2·1=2
θ→0 θ 2θ→0 2θ · 1 2θ→0 2θ
2
OR use a trigonometric identity:
sin 2θ 2 sin θ cos θ sin θ
lim = lim = 2· lim · lim cos θ = 2·1·1 = 2
θ→0 θ θ→0 θ θ→0 θ θ→0
. . . . . .
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96. Summary
Limits laws say limits play
well with the rules of
arithmetic
When limit laws do not
work we can be
algebraically creative
When algebra does not
work we can try
Squeezing.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 1.4 Calculating Limits May 18, 2010 37 / 37