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Lesson 4 - Limits at Infinity

1. The document discusses limits at infinity, including the formal definitions of limits as x approaches positive or negative infinity. 2. It provides an example of evaluating the limit of f(x)=x/(x+2) as x approaches positive and negative infinity, showing the function values approach 1 in both cases. 3. Guidelines are given for finding limits at infinity of rational functions based on the degrees of the numerator and denominator.

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106 views3 pages

Lesson 4 - Limits at Infinity

1. The document discusses limits at infinity, including the formal definitions of limits as x approaches positive or negative infinity. 2. It provides an example of evaluating the limit of f(x)=x/(x+2) as x approaches positive and negative infinity, showing the function values approach 1 in both cases. 3. Guidelines are given for finding limits at infinity of rational functions based on the degrees of the numerator and denominator.

Uploaded by

Lewis Pharmaceuticals
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Lesson 4: Limits at Infinity

Learning Objectives:

At the end of the lesson, learners will be able to:

1. Illustrate and evaluate limits at infinity

LIMITS AT INFINITY
x
f ( x )=
x+ 2
Consider a function defined by .
Let us allow x to increase and decrease without bound. The corresponding function
values appear in the table below:

x decreases without bound x increases without bound


−∞
+∞

X -10000 -1000 -100 -10 0 10 100 1000 10000


f(x) 1.0002 1.0020 1.0204 1.25 0 0.8333 0.9804 0.9980 0.9998
1
1 f(x) approaches 1 f(x) approaches 1

The table clearly suggests that the value of f(x) approaches 1 as x either increases or
decreases without bound. These limits at infinity are denoted by
x x
lim =1 lim =1
x →+∞ x+ 2 and x →−∞ x +2

The illustration above allows us to introduce the formal definition of limits at infinity.

DEFINITIONS OF LIMITS AT INFINITY:

Let L be a real number.

lim f (x )=L
DEFINITION 1: The limit of f(x) as x increases without bound is L, written x → +∞ ,
if for any ε > 0 , there exists a number M> 0 such that |f ( x )−L| < ∈ whenever x> M .
lim f ( x )=L
x → −∞

DEFINITION 2: The limit of f(x) as x decreases without bound is L, written


ε >0 N <0 |f ( x )−L| < ∈ x< N
, if for any , there exists a number such that whenever .

x → +∞
REMARKS: The symbol is used to denote that x increases without bound while
x → −∞
the symbol is used to denote that x decreases without bound.

Besides the basic theorems and corollaries on limits, as discussed in the previous
lessons, the following theorem is likewise essential in evaluating limits at infinity:

THEOREM: If r is a positive rational number and c is any real number, then

c c
lim =0 lim =0
x → +∞ xr x → −∞ xr
a) b)

GUIDELINES FOR FINDING LIMITS AT ±∞ OF RATIONAL FUNCTIONS:

1. If the degree of the numerator is less than the degree of the denominator, then the limit
of the rational function is 0.

2. If the degree of the numerator is equal to the degree of the denominator, then the limit of
the rational function is the ratio of the leading coefficients.
3. If the degree of the numerator is greater than the degree of the denominator, then the
limit of the rational function does not exist.

Definition: (HORIZONTAL ASYMPTOTE)

lim f ( x ) = L
The line y = L is a horizontal asymptote of the graph of f if x → +∞ or
lim f ( x ) = L
x → −∞ .
Note that from this definition, the graph of a function of x can have at most two horizontal
asymptotes.

Example A: Evaluate the limit of the function


5 2 x+ 3
1. lim 3 2. lim
x→+∞ x x→+∞ 4−x

2 4
x +2 x +1 x +1
3. lim 4. lim
x→−∞ x 3+ 4 x→−∞ x 2+3 x

x
5. lim
x→+∞ √ x 2−1

Example B: Determine the horizontal asymptote(s) of the graph of the function.

1.
f ( x )=
x +1
x−4 2.
( )
g ( x ) =4 1−
1
x2

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