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AP Calculus BC
Unit 1: Limits and Continuity
1.14 Connecting Infinite Limits and Vertical Asymptotes
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What is a vertical asymptote defined as?
Vertical line
x
=
x =
x
=
c
c
c
Infinite limits occur when the value of
f
(
x
)
f(x)
f
(
x
)
approaches \pm \infty
The function
f
(
x
)
=
f(x) =
f
(
x
)
=
1
(
x
−
2
)
2
\frac{1}{(x - 2)^{2}}
(
x
−
2
)
2
1
has a vertical asymptote at
x
=
x =
x
=
2
2
2
What is a key characteristic of infinite limits that indicates the presence of a vertical asymptote?
Approaching infinity near
x
=
x =
x
=
c
c
c
Match the feature with its description:
Infinite Limits ↔️ Values of
f
(
x
)
f(x)
f
(
x
)
increase or decrease without bound as
x
x
x
approaches a value.
Vertical Asymptotes ↔️ Line
x
=
x =
x
=
c
c
c
where the function approaches infinity.
Graph of Infinite Limits ↔️ Function values grow unboundedly near
x
=
x =
x
=
c
c
c
Graph of Vertical Asymptotes ↔️ Vertical line where the function tends to infinity.
What does an infinite limit indicate about a function near a specific point?
Function values grow unboundedly
The function f(x) = \frac{1}{x - 3}</latex> has a vertical asymptote at
x
=
x =
x
=
3
3
3
Infinite limits lead to a vertical asymptote, which is a vertical line where the function tends to
infinity
The vertical asymptote of f(x) = \frac{1}{x - 2}</latex> is
x
=
x =
x
=
2
2
2
What does an infinite limit lead to in a function?
Vertical asymptote
Infinite limits occur when a function approaches
+
+
+
∞
\infty
∞
or
−
∞
- \infty
−
∞
as
x
x
x
approaches a certain value.
An infinite limit leads to a
vertical
asymptote.
Match the concept with its description:
Infinite Limits ↔️ Function values increase or decrease without bound
Vertical Asymptotes ↔️ Vertical line where function tends to infinity
What is the mathematical representation of a vertical asymptote?
x
=
x =
x
=
c
c
c
Infinite limits can be represented as
lim
x
→
c
f
(
x
)
=
\lim_{x \to c} f(x) =
lim
x
→
c
f
(
x
)
=
±
∞
\pm\infty
±
∞
.
Steps to identify a vertical asymptote:
1️⃣ Check for infinite limits
2️⃣ Determine the value of
c
c
c
where
f
(
x
)
f(x)
f
(
x
)
approaches infinity
3️⃣ Verify that
lim
x
→
c
f
(
x
)
=
\lim_{x \to c} f(x) =
lim
x
→
c
f
(
x
)
=
±
∞
\pm\infty
±
∞
4️⃣ Identify the vertical asymptote as
x
=
x =
x
=
c
c
c
For the function
f
(
x
)
=
f(x) =
f
(
x
)
=
1
x
−
2
\frac{1}{x - 2}
x
−
2
1
, what is the vertical asymptote?
x = 2</latex>
What type of function has a vertical asymptote as
x
x
x
approaches 0 from the right?
Logarithmic
Match the concept with its feature:
Infinite Limits ↔️ Values of
f
(
x
)
f(x)
f
(
x
)
increase or decrease without bound
Vertical Asymptotes ↔️ Line
x
=
x =
x
=
c
c
c
where the function approaches infinity
For the function
f
(
x
)
=
f(x) =
f
(
x
)
=
1
x
−
2
\frac{1}{x - 2}
x
−
2
1
, what happens to the function as
x
x
x
approaches 2?
Approaches infinity
Infinite limits occur when a function approaches
+
+
+
∞
\infty
∞
or
−
∞
- \infty
−
∞
as
x
x
x
approaches a certain value.
In finite limits, the function approaches a specific number as
x
x
x
approaches
c
c
c
.
For the function
f
(
x
)
=
f(x) =
f
(
x
)
=
1
(
x
−
3
)
2
\frac{1}{(x - 3)^{2}}
(
x
−
3
)
2
1
, what is the infinite limit as
x
x
x
approaches 3?
∞
\infty
∞
What does the notation
lim
x
→
c
f
(
x
)
=
\lim_{x \to c} f(x) =
lim
x
→
c
f
(
x
)
=
L
L
L
signify?
A finite limit
Infinite limits occur when the function
f
(
x
)
f(x)
f
(
x
)
approaches
+
+
+
∞
\infty
∞
or
−
∞
- \infty
−
∞
as
x
x
x
approaches a certain value
Finite limits occur when
f
(
x
)
f(x)
f
(
x
)
approaches a specific number
L
L
L
as
x
x
x
approaches
c
c
c
.
Match the concept with its description:
Infinite Limits ↔️ Function values increase or decrease without bound
Vertical Asymptotes ↔️ Line
x
=
x =
x
=
c
c
c
where function approaches infinity
Infinite limits lead to
vertical asymptotes
.
Match the concept with its description:
Infinite Limits ↔️ Function values increase or decrease without bound
Vertical Asymptotes ↔️ Vertical line where function tends to infinity
The logarithmic function
f
(
x
)
=
f(x) =
f
(
x
)
=
ln
(
x
)
\ln(x)
ln
(
x
)
has a vertical asymptote as
x
x
x
approaches 0 from the right.
The exponential function
f
(
x
)
=
f(x) =
f
(
x
)
=
e
1
x
e^{\frac{1}{x}}
e
x
1
has a vertical asymptote as
x
x
x
approaches 0.
Steps to find vertical asymptotes graphically:
1️⃣ Identify potential locations where
f
(
x
)
→
±
∞
f(x) \to \pm \infty
f
(
x
)
→
±
∞
2️⃣ Check limits as
x
→
c
x \to c
x
→
c
3️⃣ Draw vertical asymptotes at
x
=
x =
x
=
c
c
c
Steps to find vertical asymptotes algebraically:
1️⃣ Identify values where the denominator is zero
2️⃣ Simplify the function
3️⃣ Check one-sided limits
The function f(x) = \frac{1}{x - 3}</latex> has a vertical asymptote at
x
=
x =
x
=
3
3
3
.
Match the feature with its description:
Infinite Limits ↔️ Values of
f
(
x
)
f(x)
f
(
x
)
increase or decrease without bound as
x
x
x
approaches a certain value.
Vertical Asymptotes ↔️ Line
x
=
x =
x
=
c
c
c
where the function approaches infinity.
What is the definition of infinite limits as
x
x
x
approaches a certain value?
Values of
f
(
x
)
f(x)
f
(
x
)
increase or decrease without bound
Infinite limits are represented using the notation
lim
x
→
c
f
(
x
)
=
\lim_{x \to c} f(x) =
lim
x
→
c
f
(
x
)
=
±
∞
\pm \infty
±
∞
, indicating that
f
(
x
)
f(x)
f
(
x
)
becomes unbounded
Infinite limits on a graph are depicted by function values growing unboundedly near
x
=
x =
x
=
c
c
c
.
What is a vertical asymptote for
f
(
x
)
=
f(x) =
f
(
x
)
=
1
x
−
2
\frac{1}{x - 2}
x
−
2
1
?
x
=
x =
x
=
2
2
2
Infinite limits occur when
f
(
x
)
f(x)
f
(
x
)
approaches
±
∞
\pm \infty
±
∞
as
x
x
x
approaches a certain value
c
c
c
.
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