Limits of Polynomial, Rational, and Radical
Functions
by CHED on June 17, 2017
lesson duration of 3 minutes
under Basic Calculus
generated on June 17, 2017 at 10:58 pm
Tags: The Limit of a Function: Theorems and Examples, Limits and Continuity
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Generated: Jun 18,2017 06:58 AM
Limits of Polynomial, Rational, and Radical Functions
( 3 mins )
Written By: CHED on June 21, 2016
Subjects: Basic Calculus
Tags: The Limit of a Function: Theorems and Examples, Limits and Continuity
Resources
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Content Standard
The learners demonstrate an understanding of the basic concepts of limit and continuity of a function
Performance Standard
The learners shall be able to formulate and solve accurately real-life problems involving continuity of functions
Learning Competencies
Apply the limit laws in evaluating the limit of algebraic functions (polynomial, rational, and radical)
Introduction 1 mins
In the previous lesson, we presented and illustrated the limit theorems. We start by recalling these limit theorems.
Theorem 1.
1. Let c, k, L and M be real numbers, and let f(x) and g(x) be functions defined on some open interval
containing c, except possibly at c.
1. If
exists, then it is unique. That is, if
and
then L=M.
2.
3.
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4. Suppose
and
i. (Constant Multiple)
ii. (Addition)
iii. (Multiplication)
iv. (Division)
v. (Power)
provided
for p, a positive integer.
vi. (Root/ Radical)
for positive integers n, and provided that L > 0 when n is even.
In this lesson, we will show how these limit theorems are used in evaluating algebraic functions. Particularly, we will
illustrate how to use them to evaluate the limits of polynomial, rational and radical functions.
Lesson Proper 1 mins
LIMITS OF ALGEBRAIC FUNCTIONS
We start with evaluating the limits of polynomial functions.
EXAMPLE 1: Determine
Solution.
Solution. From the theorems above,
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EXAMPLE 2: Determine
Solution. From the theorems above,
EXAMPLE 3:
3: Evaluate
Solution. From the theorems above,
We will now apply the limit theorems in evaluating rational functions. In evaluating the limits of such functions, recall
from Theorem 1 the Division Rule, and all the rules stated in Theorem 1 which have been useful in evaluating limits of
polynomial functions, such as the Addition and Product Rules.
EXAMPLE 4: Evaluate
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Solution. First, note that
Thus,
Since the limit of the denominator is nonzero, we can apply the Division Rule.
EXAMPLE 5: Evaluate
Solution. We start by checking the limit of the polynomial function in the denominator.
Since the limit of the denominator is not zero, it follows that
EXAMPLE 6: Evaluate
First note that
Thus, using the theorem,
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Theorem 2.Let
2.Let f be a polynomial of the for
If c is a real number, then
Proof.
Proof. Let c be any real number. Remember that a polynomial is defined at any real number. So,
Now apply the limit theorems in evaluating
Therefore,
EXAMPLE 7: Evaluate
Solution. Note first that our function
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is a polynomial. Computing for the value of f at x = ?1, we get
Therefore, from Theorem 2,
Note that we get the same answer when we use limit theorems.
Theorem 3.Let
3.Let h be a rational function of the form
real number and
where f and g are polynomial functions. If c is a
then
Proof. From Theorem 2,
Therefore, by the Division Rule of Theorem 1,
which is nonzero by assumption. Moreover,
EXAMPLE 8: Evaluate
Solution.
Solution. Since the denominator is not zero when evaluated at x = 1, we may apply
Theorem 3:
We will now evaluate limits of radical functions using limit theorems.
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EXAMPLE 9: Evaluate
Solution. Note that
Therefore, by the Radical/Root Rule,
EXAMPLE 10: Evaluate
Solution. Note that
Hence, by the Radical/Root Rule,
EXAMPLE 11:
11: Evaluate
Solution. Since the index of the radical sign is odd, we do not have to worry that the limit of the radicand is negative.
Therefore, the Radical/Root Rule implies that
EXAMPLE 12:
12: Evaluate
Solution.First,
Solution.First, note that
and Radical Rules of Theorem 1, we obtain
Moreover,
Thus, using the Division
INTUITIVE NOTIONS OF INFINITE LIMITS
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We
limit that
at a Theorem
point c of 3a does
rational
of the
form it assumes
where that
f and
g are
polynomial
with
andinvestigate theNote
notfunction
cover this
because
the
denominator
is functions
nonzero at
c.
Now, consider the function
Note that the function is not defined at x = 0 but we can check the
behavior of the function as x approaches 0 intuitively. We first consider approaching 0 from the left.
Observe that as x approaches 0 from the left, the value of the function increases without bound. When this happens,
we say that the limit of f(
f(x) as x approaches 0 from the left is positive infinity,
infinity, that is,
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Again, as x approaches 0 from the right, the value of the function increases without bound, so,
Since
and
we may conclude that
Now, consider the function
Note that the function is not defined at x = 0 but we can still check the
behavior of the function as x approaches 0 intuitively. We first consider approaching 0 from the left.
This time, as x approaches 0 from the left, the value of the function decreases without bound. So, we say that the limit
of f(x) as x approaches 0 from the left is negative infinity, that is,
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As x approaches 0 from the right, the value of the function also decreases without bound, that is,
Since
and
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