1
© MathTutorDVD.com
Calculus 1 - Limits
Worksheet 13 –
Continuity
2
© MathTutorDVD.com
Calculus 1 - Limits - Worksheet 13 – Continuity
1. Is the function
continuous at ? Explain your reasoning.
2. Is the function
continuous at ? Explain your
reasoning.
3
© MathTutorDVD.com
3. Is the function
a continuous function? Explain your reasoning.
4. Is the function
continuous at ? Explain your
reasoning.
4
© MathTutorDVD.com
5. Is the function
a continuous function? Explain your
reasoning.
6. Is the function
continuous at ? Explain your
reasoning.
5
© MathTutorDVD.com
7. Is the function
a continuous function? Explain your reasoning.
8. Is the function
continuous at ? Explain your
reasoning.
6
© MathTutorDVD.com
9. Given the functions
and
. Is
a continuous function? Explain your reasoning.
10. Is the function
continuous at ? Explain your
reasoning.
7
© MathTutorDVD.com
11. Is the function
a continuous function? Explain your
reasoning.
12. Is the function
a continuous function? Explain your
reasoning.
8
© MathTutorDVD.com
Answers - Calculus 1 - Limits - Worksheet 13 – Continuity
1. Is the function
continuous at ? Explain your reasoning.
Solution:
A function is continuous at if
Therefore,
Since is not defined at , is not continuous at
Answer: is not continuous at . because
, and
.
2. Is the function
continuous at ? Explain your
reasoning.
Solution:
A function is continuous at if
Therefore,
does not exist at , so the function cannot be continuous
at
Answer:
is not continuous at because
does not exist at
9
© MathTutorDVD.com
3. Is the function
a continuous function? Explain your reasoning.
Solution:
A function is continuous if it is continuous over its entire domain. The
denominator of the function is
with a domain of , however since
the domain of is the function is defined over its domain. The
function
is continuous on this domain, the function
is also
continuous on this domain, and the function
is continuous on this
domain. Lastly, the function
is continuous because it is a
composition of continuous functions.
Answer:
is a continuous function because it is a composition of
continuous functions.
4. Is the function
continuous at ? Explain your
reasoning.
Solution:
A function is continuous at if
First of all, based on the definition of , the function does not exist at so
it is not continuous. Furthermore, does not exist. Therefore, the function
cannot be continuous at
Therefore,
does not exist at , so the function cannot be continuous
at
Answer:
is not continuous at because
does not exist at
and the function does not exist at Either of these reasons are sufficient.
10
© MathTutorDVD.com
5. Is the function
a continuous function? Explain your
reasoning.
Solution:
The function
is a continuous function because the absolute value
function is continuous. The function
is a continuous function
because it is a polynomial function and all polynomials are continuous. Then, the
function
is a composition of two continuous functions, and
is therefore continuous.
Answer:
is a continuous function because it is a
composition of two continuous functions.
6. Is the function
continuous at ? Explain your
reasoning.
Solution:
A function is continuous at if
The first portion of the given function can be simplified after factoring:
Therefore,
. However,
, which means
.
Thus, is not continuous at
Answer: is not continuous at because
11
© MathTutorDVD.com
7. Is the function
a continuous function? Explain your reasoning.
Solution:
Simplify the function after factoring.
The function has a domain of all real numbers except When
simplified, is defined as a quotient of two polynomial functions, and all
polynomial functions are continuous. The quotient of two continuous functions is
continuous, so is continuous.
Answer: is continuous, within its domain, because it is a quotient of two
polynomial functions.
8. Is the function
continuous at ? Explain your
reasoning.
Solution:
A function is continuous at if
Therefore,
. However,
, which means
.
Thus, is not continuous at
Answer: is not continuous at because
12
© MathTutorDVD.com
9. Given the functions
and
. Is
a continuous function? Explain your reasoning.
Solution:
A function is continuous at if
However, continuity
properties will help answer this question.
The function
is continuous because it is the difference of two
continuous functions,
and
. A continuity property states that the
difference of two continuous functions is continuous. Then the function
is the sum of two continuous functions,
and
. A continuity property
states that the sum of two continuous functions is continuous. Now,
is the product of two continuous functions, and , which we
just stated are continuous. A continuity property states that the product of two
continuous functions is continuous. Therefore,
is continuous.
Answer:
is continuous because it is the product of two continuous
functions.
10. Is the function
continuous at ? Explain your
reasoning.
Solution:
A function is continuous at if
Therefore,
. Based on the definition of the function,
which
means
.
13
© MathTutorDVD.com
Answer: is continuous at because
11. Is the function
a continuous function? Explain your
reasoning.
Solution:
The function
can be simplified after factoring.
The factoring shows that and are excluded from the domain of
. Now, the function is the quotient of two polynomial functions (with a
restrictive domain). Therefore is continuous because polynomial functions
are continuous, and the quotient of two continuous functions is continuous.
Answer: is continuous, within its domain, because it is a quotient of two
continuous functions.
12. Is the function
a continuous function? Explain your
reasoning.
The function
can be simplified after factoring.
The factoring shows that is excluded from the domain of . Now, the
function is the quotient of two polynomial functions (with a restrictive
domain). Therefore is continuous because polynomial functions are
continuous, and the quotient of two continuous functions is continuous.
Answer: is continuous, within its domain, because it is a quotient of two
continuous functions.
