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AP Calculus BC
Unit 1: Limits and Continuity
1.9 Connecting Multiple Representations of Limits
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What does the limit of a function represent as the input approaches a certain point?
The value it approaches
For the function
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
, as
x
x
x
approaches 2,
f
(
x
)
f(x)
f
(
x
)
approaches 4.
Limits can be expressed using algebraic, graphical, and
tabular
methods.
Match the representation of limits with its description:
Algebraic ↔️ Expresses limit using equation
Graphical ↔️ Visual interpretation
Tabular ↔️ Uses values to infer limit
Arrange the different representations of limits from most abstract to most concrete:
1️⃣ Algebraic
2️⃣ Graphical
3️⃣ Tabular
What is the limit of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
as
x
x
x
approaches 2?
4
Graphical representation of limits involves observing the function's behavior as
x
x
x
approaches a certain point on a graph.
The tabular representation of limits involves providing values of
x
x
x
near
a
a
a
and their corresponding
f
(
x
)
f(x)
f
(
x
)
values to infer the limit.
What does the algebraic expression \lim_{x \to a} f(x) = L</latex> mean in the context of limits?
As
x
x
x
approaches
a
a
a
,
f
(
x
)
f(x)
f
(
x
)
approaches
L
L
L
What does the graphical representation of a limit show visually?
The y-value the function approaches
For the function f(x) = \frac{x^{2} - 4}{x - 2}</latex>, as
x
x
x
approaches 2,
f
(
x
)
f(x)
f
(
x
)
approaches 4 based on tabular values.
What is the limit of a function at
x
=
x =
x
=
a
a
a
?
The y-value approached
We can infer a limit by providing values of
x
x
x
near
a
a
a
and their corresponding
f
(
x
)
f(x)
f
(
x
)
values in a tabular representation.
The algebraic representation of a limit uses an
equation
.
Match the values of
x
x
x
near 2 with their corresponding
f
(
x
)
f(x)
f
(
x
)
values:
1.9 ↔️ 3.9
1.99 ↔️ 3.99
2.01 ↔️ 4.01
2.1 ↔️ 4.1
What does the symbolic representation
lim
x
→
a
f
(
x
)
=
\lim_{x \to a} f(x) =
lim
x
→
a
f
(
x
)
=
L
L
L
mean?
As x approaches a, f(x) approaches L
In a graphical representation, the limit is the y-value the function approaches at
x
=
x =
x
=
a
a
a
.
The limit of
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
as
x
→
2
x \to 2
x
→
2
is 4.
What is the symbolic representation of the limit of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x^{2} - 4
x
2
−
4
as x \to 3</latex>?
lim
x
→
3
(
x
2
−
4
)
\lim_{x \to 3} (x^{2} - 4)
lim
x
→
3
(
x
2
−
4
)
Evaluating the limit of
lim
x
→
3
(
x
2
−
4
)
\lim_{x \to 3} (x^{2} - 4)
lim
x
→
3
(
x
2
−
4
)
, we find the value is 5.
The graph of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x^{2} - 4
x
2
−
4
approaches
y
=
y =
y
=
5
5
5
as
x
→
3
x \to 3
x
→
3
.
Match the feature with its representation:
Interpretation ↔️ Behavior of function near
x
=
x =
x
=
a
a
a
Representation ↔️ Graph of function
Visualization ↔️ Visual inspection of trend
Expression ↔️
lim
x
→
a
f
(
x
)
=
\lim_{x \to a} f(x) =
lim
x
→
a
f
(
x
)
=
L
L
L
What does the symbolic representation
lim
x
→
a
f
(
x
)
=
\lim_{x \to a} f(x) =
lim
x
→
a
f
(
x
)
=
L
L
L
indicate in words?
As x approaches a, f(x) approaches L
Graphically, the limit is the y-value that the function approaches as
x
x
x
approaches
a
a
a
.
The graph of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
approaches y =4</latex> as
x
→
2
x \to 2
x
→
2
.
What does the symbolic representation
lim
x
→
2
x
2
−
4
x
−
2
=
\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} =
lim
x
→
2
x
−
2
x
2
−
4
=
4
4
4
mean graphically?
The graph approaches y = 4 as x approaches 2
The graphical interpretation of
lim
x
→
2
x
2
−
4
x
−
2
=
\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} =
lim
x
→
2
x
−
2
x
2
−
4
=
4
4
4
is that the graph approaches
y
=
y =
y
=
4
4
4
as
x
x
x
approaches 2.
The symbolic representation
lim
x
→
2
x
2
−
4
x
−
2
=
\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} =
lim
x
→
2
x
−
2
x
2
−
4
=
4
4
4
corresponds to a graph that approaches
y
=
y =
y
=
4
4
4
as
x
x
x
approaches
2
2
2
.
What y-value does the graph of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
approach as
x
x
x
approaches 2?
4
Match the representation of limits with its description:
Symbolic ↔️ Equation using
lim
x
→
a
\lim_{x \to a}
lim
x
→
a
Graphical ↔️ y-value approached on a graph
What are the three main representations of limits described in the material?
Algebraic, graphical, tabular
Match the representation of limits with its example:
Algebraic ↔️
lim
x
→
2
x
2
−
4
x
−
2
=
\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} =
lim
x
→
2
x
−
2
x
2
−
4
=
4
4
4
Graphical ↔️ A graph where the function approaches
y
=
y =
y
=
4
4
4
as
x
→
2
x \to 2
x
→
2
Tabular ↔️
x
=
x =
x
=
1.9
,
1.99
,
2.01
,
2.1
;
f
(
x
)
≈
3.9
,
3.99
,
4.01
,
4.1
1.9, 1.99, 2.01, 2.1; f(x) \approx 3.9, 3.99, 4.01, 4.1
1.9
,
1.99
,
2.01
,
2.1
;
f
(
x
)
≈
3.9
,
3.99
,
4.01
,
4.1
What is the value of
lim
x
→
2
x
2
−
4
x
−
2
\lim_{x \to 2} \frac{x^{2} - 4}{x - 2}
lim
x
→
2
x
−
2
x
2
−
4
?
4
The graphical representation of a limit shows the y-value the function approaches as x approaches a
specific
value.
The tabular representation of a limit uses values of
x
x
x
near
a
a
a
to infer the limit by observing the behavior of
f
(
x
)
f(x)
f
(
x
)
values.converge
Match the representation of a limit with its description:
Algebraic ↔️ Uses equation
Graphical ↔️ Visual interpretation
Tabular ↔️ Uses values to infer limit
What value does
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
approach as
x
x
x
approaches 2?
4
The graphical representation of a limit is the y-value the function
approaches
on its graph as x gets close to a from both sides.
For
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x^{2} - 4
x
2
−
4
, as
x
x
x
approaches 3, the graph approaches the height
y
=
y =
y
=
5
5
5
, illustrating the graphical representation of the limit
Steps to infer a limit from tabular data:
1️⃣ Observe the values of
f
(
x
)
f(x)
f
(
x
)
as
x
x
x
approaches
a
a
a
2️⃣ Check if the values of
f
(
x
)
f(x)
f
(
x
)
converge to a number
L
L
L
3️⃣ Conclude that
lim
x
→
a
f
(
x
)
=
\lim_{x \to a} f(x) =
lim
x
→
a
f
(
x
)
=
L
L
L
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