Save
AP Calculus BC
Unit 1: Limits and Continuity
1.4 Estimating Limit Values from Tables
Save
Share
Learn
Content
Leaderboard
Share
Learn
Cards (60)
What does a limit tell us about a function's behavior?
Value as input approaches
As
x
x
x
approaches 2,
f
(
x
)
f(x)
f
(
x
)
approaches 4
The limit of
f
(
x
)
f(x)
f
(
x
)
as
x
x
x
approaches 2 is written as
lim
x
→
2
f
(
x
)
=
\lim_{x \to 2} f(x) =
lim
x
→
2
f
(
x
)
=
4
4
4
.
What is the function
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
equal to when
x
=
x =
x
=
0
0
0
?
2
Steps to estimate a limit from a table
1️⃣ Examine values from the left
2️⃣ Examine values from the right
3️⃣ Determine the common approach
What do we examine to estimate limits?
Function values
To estimate limits, we examine the values of a function as it approaches a target point from both the left and the
right
Estimating limits involves examining function values as
x
x
x
approaches a target point from only one direction.
False
What value does
f
(
x
)
f(x)
f
(
x
)
approach as
x
x
x
approaches 2 from the left in the given table?
4
As
x
x
x
approaches 2 from the right,
f
(
x
)
f(x)
f
(
x
)
approaches the value of 4
From the given table, \lim_{x \to 2} f(x) =
4
</latex>.
Steps to estimate the limit of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
as
x
x
x
approaches 0.
1️⃣ Choose values of
x
x
x
near 0 from the left and right
2️⃣ Evaluate
f
(
x
)
f(x)
f
(
x
)
for these
x
x
x
values
3️⃣ Observe the trend in
f
(
x
)
f(x)
f
(
x
)
4️⃣ Conclude the limit value
What is the function given in the example for estimating limits?
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
What do we examine to estimate limits of a function?
Function values from left and right
As
x
x
x
approaches 2 from the left,
f
(
x
)
f(x)
f
(
x
)
approaches 4
If
f
(
x
)
f(x)
f
(
x
)
approaches 4 as
x
x
x
approaches 2 from both sides, then \lim_{x \to 2} f(x) = 4</latex>.
For
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
, what value does
f
(
x
)
f(x)
f
(
x
)
approach as
x
x
x
approaches 0?
2
What two directions do we examine to estimate limits?
Left and right
As
x
x
x
approaches 2 from the left,
f
(
x
)
f(x)
f
(
x
)
approaches 4
What is the function used in the example to estimate the limit as
x
x
x
approaches 0?
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
To estimate limits, we only need to examine values approaching from the left.
False
As
x
x
x
approaches 2 from the right,
f
(
x
)
f(x)
f
(
x
)
approaches 4
What is the value of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
when
x
=
x =
x
=
−
0.1
- 0.1
−
0.1
?
2.01
2.01
2.01
We can estimate limits by examining values of a function as it approaches a target point from both the left and
right
.
Match the value of
x
x
x
with the corresponding value of
f
(
x
)
f(x)
f
(
x
)
when
lim
x
→
2
f
(
x
)
=
\lim_{x \to 2} f(x) =
lim
x
→
2
f
(
x
)
=
4
4
4
:
1.9
1.9
1.9
↔️
3.8
3.8
3.8
1.99
1.99
1.99
↔️
3.98
3.98
3.98
2.01
2.01
2.01
↔️
4.02
4.02
4.02
2.1
2.1
2.1
↔️
4.2
4.2
4.2
What is the limit of f(x) = x^{2} + 2</latex> as
x
x
x
approaches 0?
lim
x
→
0
f
(
x
)
=
\lim_{x \to 0} f(x) =
lim
x
→
0
f
(
x
)
=
2
2
2
Estimating limits involves analyzing the behavior of a function as it approaches a specific point from both the left and
right
.
To estimate limits, we examine the values of a function as it approaches a target point from the left and
right
Match the value of
x
x
x
with the corresponding value of
f
(
x
)
f(x)
f
(
x
)
when
lim
x
→
0
f
(
x
)
=
\lim_{x \to 0} f(x) =
lim
x
→
0
f
(
x
)
=
2
2
2
:
−
0.1
- 0.1
−
0.1
↔️
2.01
2.01
2.01
−
0.01
- 0.01
−
0.01
↔️
2.0001
2.0001
2.0001
0
0
0
↔️
2
2
2
How does the example function
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
behave as
x
x
x
approaches 0 from the left?
Approaches 2
Estimating limits requires examining only the values of
f
(
x
)
f(x)
f
(
x
)
as
x
x
x
approaches the target point from the left.
False
As
x
x
x
approaches 2 from the left,
f
(
x
)
f(x)
f
(
x
)
approaches 4
Order the steps to estimate a limit using values approaching from the left and right.
1️⃣ Choose a target point
x
x
x
.
2️⃣ Examine values of
f
(
x
)
f(x)
f
(
x
)
as
x
x
x
approaches from the left.
3️⃣ Examine values of
f
(
x
)
f(x)
f
(
x
)
as
x
x
x
approaches from the right.
4️⃣ Determine the value
f
(
x
)
f(x)
f
(
x
)
approaches from both sides.
What is the limit of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
as
x
x
x
approaches 0 from the right?
lim
x
→
0
f
(
x
)
=
\lim_{x \to 0} f(x) =
lim
x
→
0
f
(
x
)
=
2
2
2
To estimate limits, we must always create a table of values to analyze.
False
As
x
x
x
approaches 2 from the right,
f
(
x
)
f(x)
f
(
x
)
approaches 4
What values of
x
x
x
are used to estimate the limit of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
as
x
x
x
approaches 0?
−
0.1
- 0.1
−
0.1
,
−
0.01
- 0.01
−
0.01
, 0
What value does
f
(
x
)
f(x)
f
(
x
)
approach as
x
x
x
approaches 2 from the left?
4
What is the function given as an example to estimate a limit?
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
+
x^{2} +
x
2
+
2
2
2
For f(x) = x^{2} + 2</latex>, as
x
x
x
approaches 0,
f
(
x
)
f(x)
f
(
x
)
approaches 2
See all 60 cards
See similar decks
1.4 Estimating Limit Values from Tables
AP Calculus AB > Unit 1: Limits and Continuity
40 cards
1.3 Estimating Limit Values from Graphs
AP Calculus BC > Unit 1: Limits and Continuity
111 cards
1.3 Estimating Limit Values from Graphs
AP Calculus AB > Unit 1: Limits and Continuity
32 cards
1.2 Defining Limits and Using Limit Notation
AP Calculus AB > Unit 1: Limits and Continuity
70 cards
1.2 Defining Limits and Using Limit Notation
AP Calculus BC > Unit 1: Limits and Continuity
86 cards
Unit 1: Limits and Continuity
AP Calculus AB
748 cards
Unit 1: Limits and Continuity
AP Calculus BC
1405 cards
1.5 Determining Limits Using Algebraic Properties
AP Calculus AB > Unit 1: Limits and Continuity
106 cards
1.9 Connecting Multiple Representations of Limits
AP Calculus BC > Unit 1: Limits and Continuity
186 cards
1.5 Determining Limits Using Algebraic Properties
AP Calculus BC > Unit 1: Limits and Continuity
106 cards
1.8 Determining Limits Using the Squeeze Theorem
AP Calculus AB > Unit 1: Limits and Continuity
70 cards
1.14 Connecting Infinite Limits and Vertical Asymptotes
AP Calculus BC > Unit 1: Limits and Continuity
51 cards
1.8 Determining Limits Using the Squeeze Theorem
AP Calculus BC > Unit 1: Limits and Continuity
43 cards
1.9 Connecting Multiple Representations of Limits
AP Calculus AB > Unit 1: Limits and Continuity
74 cards
1.6 Determining Limits Using Algebraic Manipulation
AP Calculus BC > Unit 1: Limits and Continuity
38 cards
1.14 Connecting Infinite Limits and Vertical Asymptotes
AP Calculus AB > Unit 1: Limits and Continuity
34 cards
1.7 Selecting Procedures for Determining Limits
AP Calculus BC > Unit 1: Limits and Continuity
209 cards
1.11 Defining Continuity at a Point
AP Calculus AB > Unit 1: Limits and Continuity
42 cards
1.12 Confirming Continuity over an Interval
AP Calculus AB > Unit 1: Limits and Continuity
51 cards
1.6 Determining Limits Using Algebraic Manipulation
AP Calculus AB > Unit 1: Limits and Continuity
54 cards
1.7 Selecting Procedures for Determining Limits
AP Calculus AB > Unit 1: Limits and Continuity
48 cards