Save
AP Calculus BC
Unit 1: Limits and Continuity
1.11 Defining Continuity at a Point
Save
Share
Learn
Content
Leaderboard
Share
Learn
Cards (211)
What does the limit of a function
f
(
x
)
f(x)
f
(
x
)
as
x
x
x
approaches
a
a
a
represent?
The value
f
(
x
)
f(x)
f
(
x
)
gets closer to as
x
x
x
approaches
a
a
a
To calculate a limit, you always substitute the value
a
a
a
directly into
f
(
x
)
f(x)
f
(
x
)
.
False
What are the three conditions for continuity of a function
f
(
x
)
f(x)
f
(
x
)
at
x
=
x =
x
=
a
a
a
?
Function exists, limit exists, value equals limit
Steps to check continuity of
f
(
x
)
f(x)
f
(
x
)
at
x
=
x =
x
=
2
2
2
1️⃣
f
(
2
)
f(2)
f
(
2
)
is defined
2️⃣
lim
x
→
2
f
(
x
)
\lim_{x \to 2} f(x)
lim
x
→
2
f
(
x
)
exists
3️⃣
f
(
2
)
=
f(2) =
f
(
2
)
=
lim
x
→
2
f
(
x
)
\lim_{x \to 2} f(x)
lim
x
→
2
f
(
x
)
Consider the function
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
for
x
≠
2
x \neq 2
x
=
2
and
f
(
2
)
=
f(2) =
f
(
2
)
=
4
4
4
. Is
f
(
x
)
f(x)
f
(
x
)
continuous at
x
=
x =
x
=
2
2
2
?
Yes
For a function
f
(
x
)
f(x)
f
(
x
)
to be continuous at
x
=
x =
x
=
a
a
a
,
f
(
a
)
f(a)
f
(
a
)
must be defined
The limit of
f
(
x
)
f(x)
f
(
x
)
as
x
x
x
approaches
a
a
a
can always be calculated by direct substitution.
False
What does it mean for a function
f
(
x
)
f(x)
f
(
x
)
to be continuous at a point
x
=
x =
x
=
a
a
a
?
It satisfies three continuity conditions
For continuity at
x
=
x =
x
=
a
a
a
,
lim
x
→
a
f
(
x
)
\lim_{x \to a} f(x)
lim
x
→
a
f
(
x
)
must exist
The function
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
is continuous at
x
=
x =
x
=
2
2
2
if
f
(
2
)
=
f(2) =
f
(
2
)
=
4
4
4
.
What is the third condition for continuity at x = a</latex>?
f
(
a
)
=
f(a) =
f
(
a
)
=
lim
x
→
a
f
(
x
)
\lim_{x \to a} f(x)
lim
x
→
a
f
(
x
)
For
lim
x
→
a
f
(
x
)
\lim_{x \to a} f(x)
lim
x
→
a
f
(
x
)
to exist, both the left-hand limit and right-hand limit must be equal
If a function
f
(
x
)
f(x)
f
(
x
)
satisfies all three conditions for continuity at
x
=
x =
x
=
a
a
a
, then it is continuous at that point.
What are the three conditions for continuity of a function
f
(
x
)
f(x)
f
(
x
)
at a point
x
=
x =
x
=
a
a
a
?
Function exists, limit exists, value equals limit
For continuity at
x
=
x =
x
=
a
a
a
,
lim
x
→
a
f
(
x
)
\lim_{x \to a} f(x)
lim
x
→
a
f
(
x
)
must exist, meaning both the left-hand and right-hand limits must be equal
A function
f
(
x
)
f(x)
f
(
x
)
is continuous at a point
x
=
x =
x
=
a
a
a
if
f
(
a
)
f(a)
f
(
a
)
is defined
For a function to be continuous at
x
=
x =
x
=
a
a
a
, the left-hand and right-hand limits must be equal.
For a function to be continuous at
x
=
x =
x
=
a
a
a
,
f
(
a
)
f(a)
f
(
a
)
must equal \lim_{x \to a} f(x)</latex>
Steps to check continuity of
f
(
x
)
f(x)
f
(
x
)
at
x
=
x =
x
=
2
2
2
for
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
and
f
(
2
)
=
f(2) =
f
(
2
)
=
4
4
4
1️⃣
f
(
2
)
=
f(2) =
f
(
2
)
=
4
4
4
is defined
2️⃣
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
exists
3️⃣
f
(
2
)
=
f(2) =
f
(
2
)
=
lim
x
→
2
f
(
x
)
=
\lim_{x \to 2} f(x) =
lim
x
→
2
f
(
x
)
=
4
4
4
The function
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
is continuous at x =2</latex> if
f
(
2
)
=
f(2) =
f
(
2
)
=
4
4
4
.
A table summarizing continuity conditions includes checking if
lim
x
→
2
f
(
x
)
\lim_{x \to 2} f(x)
lim
x
→
2
f
(
x
)
exists
If a function is discontinuous at
x
=
x =
x
=
a
a
a
, it must fail to meet all three continuity conditions.
False
A function is discontinuous at x = a</latex> if
f
(
a
)
f(a)
f
(
a
)
is not defined
Match the type of discontinuity with its definition:
Removable ↔️ The limit exists but
f
(
a
)
f(a)
f
(
a
)
is not defined or
f
(
a
)
≠
lim
x
→
a
f
(
x
)
f(a) \neq \lim_{x \to a} f(x)
f
(
a
)
=
lim
x
→
a
f
(
x
)
Jump ↔️ The left-hand and right-hand limits exist but are not equal
Infinite ↔️ The function approaches infinity or negative infinity as
x
x
x
approaches
a
a
a
A removable discontinuity occurs when the limit exists but
f
(
a
)
f(a)
f
(
a
)
is undefined.
To calculate a limit, first substitute
a
a
a
into
f
(
x
)
f(x)
f
(
x
)
. If you encounter an indeterminate form, simplify the function
For a function to be continuous at
x
=
x =
x
=
a
a
a
, its value at
x
=
x =
x
=
a
a
a
must equal the limit as
x
x
x
approaches
a
a
a
.
The condition "Function Exists" for continuity requires that
f
(
a
)
f(a)
f
(
a
)
is defined.
The condition "Limit Exists" for continuity requires that both the left-hand and right-hand limits are
equal
The condition "Value Equals Limit" for continuity requires that
f
(
a
)
=
f(a) =
f
(
a
)
=
lim
x
→
a
f
(
x
)
\lim_{x \to a} f(x)
lim
x
→
a
f
(
x
)
.
Consider the function
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
for
x
≠
2
x \neq 2
x
=
2
and
f
(
2
)
=
f(2) =
f
(
2
)
=
4
4
4
. To check continuity at
x
=
x =
x
=
2
2
2
,
f
(
2
)
f(2)
f
(
2
)
is defined as 4
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 function
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
is continuous at
x
=
x =
x
=
2
2
2
.
Match the type of discontinuity with its definition:
Removable ↔️ The limit exists but
f
(
a
)
f(a)
f
(
a
)
is not defined or
f
(
a
)
≠
lim
x
→
a
f
(
x
)
f(a) \neq \lim_{x \to a} f(x)
f
(
a
)
=
lim
x
→
a
f
(
x
)
Jump ↔️ The left-hand and right-hand limits exist but are not equal
Infinite ↔️ The function approaches infinity or negative infinity as
x
x
x
approaches
a
a
a
Give an example of a removable discontinuity.
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
at
x
=
x =
x
=
2
2
2
A jump discontinuity occurs when the left-hand and right-hand limits exist but are
unequal
What happens to the function in an infinite discontinuity as
x
x
x
approaches
a
a
a
?
It approaches infinity
The function
f
(
x
)
=
f(x) =
f
(
x
)
=
\begin{cases} x +
1 & x < 2 \\ 2x - 1 & x \geq 2 \end{cases}
is continuous at
x
=
x =
x
=
2
2
2
.
The Intermediate Value Theorem (IVT) guarantees that a function takes on a specific value between two given
points
The Intermediate Value Theorem requires the function to be
continuous
on a closed interval.
See all 211 cards