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CHAPTER 2:
Limits and Continuity
2.1: An Introduction to Limits
2.2: Properties of Limits
2.3: Limits and Infinity I: Horizontal Asymptotes (HAs)
2.4: Limits and Infinity II: Vertical Asymptotes (VAs)
2.5: The Indeterminate Forms 0/0 and /
2.6: The Squeeze (Sandwich) Theorem
2.7: Precise Definitions of Limits
2.8: Continuity
- The conventional approach to calculus is founded on limits.
- In this chapter, we will develop the concept of a limit by example.
- Properties of limits will be established along the way.
- We will use limits to analyze asymptotic behaviors of functions and their graphs.
- Limits will be formally defined near the end of the chapter.
- Continuity of a function (at a point and on an interval) will be defined using limits.
SECTION 2.1: AN INTRODUCTION TO LIMITS
LEARNING OBJECTIVES
- Understand the concept of (and notation for) a limit of a rational function at a point in its domain, and understand that “limits are local.”
- Evaluate such limits.
- Distinguish between one-sided (left-hand and right-hand) limits and two-sided limits and what it means for such limits to exist.
- Use numerical / tabular methods to guess at limit values.
- Distinguish between limit values and function values at a point.
- Understand the use of neighborhoods and punctured neighborhoods in the evaluation of one-sided and two-sided limits.
- Evaluate some limits involving piecewise-defined functions.
PART A: THE LIMIT OF A FUNCTION AT A POINT
Our study of calculus begins with an understanding of the expression lim x a
f (^) ( x ) ,
where a is a real number (in short, a ) and f is a function. This is read as:
“the limit of f ( x ) as x approaches a .”
- WARNING 1: means “approaches.” Avoid using this symbol outside the context of limits.
- lim x a is called a limit operator. Here, it is applied to the function f.
lim x a
f (^) ( x ) is the real number that f (^) ( x ) approaches as x approaches a , if such a
number exists. If f ( x ) does, indeed, approach a real number, we denote that
number by L (for limit value). We say the limit exists , and we write:
lim x a f (^) ( x ) = L , or f (^) ( x ) L as x a.
These statements will be rigorously defined in Section 2.7.
lim x 1 f (^) ( x) = lim x 1 (^3 x^2 +^ x^ ^1 )
WARNING 3: Use grouping symbols when taking the limit of an expression consisting of more than one term.
= 3 1( )
2
WARNING 4: Do not omit the limit operator lim x 1 until this
substitution phase.
WARNING 5: When performing substitutions , be prepared to use grouping symbols. Omit them only if you are sure they are unnecessary. = 3
We can write: lim x 1 f (^) ( x ) = 3 , or f (^) ( x) 3 as x 1.
- Be prepared to work with function and variable names other than f and x.
For example, if g t ( ) = 3 t^2 + t 1 , then lim t 1 g t ( ) = 3 , also.
The graph of y = f (^) ( x ) is below.
Imagine that the arrows in the figure represent two lovers running towards each other along the parabola. What is the y -coordinate of the point they are approaching as they approach x = 1? It is 3, the limit value.
TIP 1: Remember that y -coordinates of points along the graph correspond to function values. §
Example 2 (Evaluating the Limit of a Rational Function at a Point)
Let f (^) ( x) =
2 x + 1 x 2
. Evaluate lim x 3 f (^) ( x ).
§ Solution
f is a rational function with implied domain Dom (^) ( f) = (^) { x x (^2) }. We observe that 3 is in the domain of f (^) ( in short, 3 Dom (^) ( f )), so we substitute (“plug in”) x = 3 and evaluate f (^) ( ) 3.
lim x 3
f (^) ( x) = lim x 3
2 x + 1 x 2
=
2 3( ) + 1
( )^3 ^2 = 7
The graph of y = f (^) ( x ) is below.
Note: As is often the case, you might not know how to draw the graph until later.
- Asymptotes. The dashed lines are asymptotes, which are lines that a graph approaches
- in a “long-run” sense (see the horizontal asymptote, or “HA,” at y = 2 ), or
- in an “explosive” sense (see the vertical asymptote, or “VA,” at x = 2 ).
“HA”s and “VA”s will be defined using limits in Sections 2. and 2.4, respectively.
PART B: ONE- AND TWO-SIDED LIMITS; EXISTENCE OF LIMITS
lim x a is a two-sided limit operator in lim x a f (^) ( x ) , because we must consider the
behavior of f as x approaches a from both the left and the right.
lim x a ^
is a one-sided left-hand limit operator. lim x a ^
f (^) ( x ) is read as:
“the limit of f ( x ) as x approaches a from the left .”
lim x a +^
is a one-sided right-hand limit operator. lim x a +^
f (^) ( x ) is read as:
“the limit of f ( x ) as x approaches a from the right .”
Example 4 (Using a Numerical / Tabular Approach to Guess a Left-Hand Limit Value)
Guess the value of lim x 3 ^ ( x^ +^3 ) using a^ table^ of function values.
§ Solution
Let f (^) ( x) = x + 3. lim x 3 ^
f (^) ( x) is the real number, if any, that f (^) ( x)
approaches as x approaches 3 from lesser (or lower) numbers. That is, we approach x = 3 from the left along the real number line.
We select an increasing sequence of real numbers ( x values) approaching 3 such that all the numbers are close to (but less than) 3. We evaluate the function at those numbers, and we guess the limit value, if any, the function values are approaching. For example:
x 2.9 2.99 2.999 (^) 3 f (^) ( x) = x + (^3) 5.9 5.99 5.999 ^ 6 (?)
We guess: lim x 3 ^ ( x^ +^3 ) =^6.
WARNING 6: Do not confuse superscripts with signs of numbers. Be careful about associating the “ ” superscript with negative numbers. Here, we consider positive numbers that are close to 3.
- If we were taking a limit as x approached 0 , then we would associate the “ ^ ” superscript with negative numbers and the “+” superscript with positive numbers.
The graph of y = f (^) ( x ) is below. We only consider the behavior of f “immediately” to the left of x = 3.
WARNING 7: The numerical / tabular approach is unreliable , and it is typically unacceptable as a method for evaluating limits on exams. (See Part D, Example 11 to witness a failure of this method.) However, it may help us guess at limit values, and it strengthens our understanding of limits. §
Example 5 (Using a Numerical / Tabular Approach to Guess a Right-Hand Limit Value)
Guess the value of lim x 3 +^ ( x^ +^3 ) using a^ table^ of function values.
§ Solution
Let f (^) ( x) = x + 3. lim x 3 +^
f (^) ( x) is the real number, if any, that f (^) ( x)
approaches as x approaches 3 from greater (or higher) numbers. That is, we approach x = 3 from the right along the real number line.
We select a decreasing sequence of real numbers ( x values) approaching 3 such that all the numbers are close to (but greater than) 3. We evaluate the function at those numbers, and we guess the limit value, if any, the function values are approaching. For example:
x (^) 3 +^ 3.001 3.01 3. f (^) ( x) = x + 3 6 (?) (^) 6.001 6.01 6.
We guess: lim x 3 +^ ( x^ +^3 ) =^6.
WARNING 8: Substitution might not work if f is not a rational function.
Example 6 (Pitfalls of Substituting into a Function that is Not Rational)
Let f ( x) = x + 1. Evaluate lim
x 0 +^
f ( x), lim
x 0 ^
f ( x), and lim
x 0
f ( x).
§ Solution
Observe that Dom ( f ) = { x x 0 } = 0, ), because x is real when
x 0 , but it is not real when x < 0.
This is important, because x is only allowed to approach 0 (or whatever a is)
through Dom ( f ). Here, x is allowed to approach 0 from the right but not
from the left.
Right-Hand Limit: lim x 0 +^
f ( x ) = 1.
Substituting x = 0 works: lim x 0 +^
f ( x) = lim
x 0 +^
( x^ +^1 ) =^0 +^1 =^1.
Left-Hand Limit: lim x 0 ^
f ( x ) does not exist (DNE).
Substituting x = 0 does not work here.
Two-Sided Limit: lim x 0
f ( x) does not exist (DNE).
This is because the corresponding left-hand limit does not exist (DNE).
Observe that f is not a rational function, so the aforementioned theorem
does not apply, even though 0 Dom ( f ). f is, however, an algebraic
function, and we will discuss algebraic functions in Section 2.2. §
PART C: IGNORING THE FUNCTION AT a
Example 7 (Ignoring the Function at ‘a’ When Evaluating a Limit; Modifying Examples 4 and 5)
Let g x ( ) = x + 3, (^) ( x (^3) ).
(We are deleting 3 from the domain of the function in Examples 4 and 5; this changes the function.)
Evaluate lim x 3 ^
g x ( ) , lim x 3 +^
g x ( ) , and lim x 3 g x ( ).
§ Solution
Since 3 Dom (^) ( g ) , we must delete the point (^) ( 3, 6) from the graph of y = x + 3 to obtain the graph of g below.
We say that g has a removable discontinuity at x = 3 (see Section 2.8), and the graph of g has a hole at the point (^) ( 3, 6).
Observe that, as x approaches 3 from the left and from the right,
g x ( ) approaches 6, even though g x ( ) never equals 6.
g (^) ( ) 3 is undefined, yet the following statements are true:
lim x 3 ^
g x ( ) = 6 ,
lim x 3 +^
g x ( ) = 6 , and
lim x 3 g x ( ) = 6.
There literally does not have to be a point at x = 3 (in general, x = a ) for
these limits to exist! Observe that substituting x = 3 into ( x + 3 ) works. §
The existence (or value) of lim x a f (^) ( x ) need not depend on the
existence (or value) of f ( a ).
- Sometimes, it does help to know what f ( a ) is when evaluating lim x a f (^) ( x ).
In Section 2.8, we will say that f is continuous at a lim x a f (^) ( x ) = f (^) ( a ) ,
provided that lim x a f (^) ( x ) and f (^) ( a ) exist. We appreciate continuity , because we
can then simply substitute x = a to evaluate a limit, which was what we did when we applied the Basic Limit Theorem for Rational Functions in Part A.
- In Examples 7 and 8, we dealt with functions that were not continuous at x = 3 ,
yet substituting x = 3 into ( x + 3 ) allowed us to evaluate the one- and two-sided
limits at a = 3. We will develop theorems that cover these Examples. We first need the following definitions.
A neighborhood of a is an open interval along the real number line that is symmetric about a.
For example, the interval (^) ( 0, 2) is a neighborhood of 1. Since 1 is the midpoint of (^) ( 0, 2) , the neighborhood is symmetric about 1.
A punctured (or deleted) neighborhood of a is constructed by taking a neighborhood of a and deleting a itself.
For example, the set (^) ( 0, 2) \ 1{ }, which can be written as (^) ( 0, 1) (^) (1, 2 ) , is a punctured neighborhood of 1. It is a set of numbers that are “immediately around” 1 on the real number line.
- The notation (^) ( 0, 2) \ 1{ } indicates that we can construct it by taking the neighborhood ( 0, 2) and deleting 1.
“Puncture Theorem” for Limits of Locally Rational Functions
Let r be a rational function, and let a Dom ( ) r.
Let f ( x ) = r x ( ) on a punctured neighborhood of x = a.
Then, lim x a
f (^) ( x ) = lim x a
r x ( ) = r a ( ).
- To evaluate the limits, substitute (“plug in”) x = a into r x ( ), and evaluate r a ( ).
- That is, if a function rule is given by a rational expression r x ( )
locally (immediately) around x^ =^ a^ , where a Dom ( ) r , then
evaluate the rational expression at a to obtain the limit of the function at a.
Refer to Examples 7 and 8. Let r x ( ) = x + 3. Observe that r is a rational function,
and 3 Dom ( ) r. Both the g and h functions were defined by x + 3 locally
(immediately) around x = 3. More precisely, they were defined by x + 3 on some
punctured neighborhood of x = 3 , say (^) (2.9, 3.1 ) \ (^) { } 3. Therefore,
lim x 3
g x ( ) = lim x 3
r x ( ) = r (^) ( ) = 3 3 + 3 = 6 , and
lim x 3 h x ( ) = lim x 3 r x ( ) = r (^) ( ) = 3 3 + 3 = 6.
It is easier to write:
lim x 3 g x ( ) = lim x 3 ( x^ +^3 ) =^3 +^3 =^6 , and
lim x 3
h x ( ) = lim x 3 ( x^ +^3 ) =^3 +^3 =^6.
The figure below refers to g , but it also applies to h. The dashed line segment at x = 3 reiterates the puncture there.
Variation of the “Puncture Theorem” for Left-Hand Limits
Let r be a rational function, and let a Dom ( ) r.
Let f ( x ) = r x ( ) on a left-neighborhood of x^ =^ a^.
Then, lim x a ^
f (^) ( x ) = lim x a ^
r x ( ) = r a ( ).
Variation of the “Puncture Theorem” for Right-Hand Limits
Let r be a rational function, and let a Dom ( ) r.
Let f ( x ) = r x ( ) on a right-neighborhood of x = a.
Then, lim x a +^
f (^) ( x ) = lim x a +^
r x ( ) = r a ( ).
Example 10 (Evaluating One-Sided and Two-Sided Limits of a Piecewise-Defined Function)
Let f (^) ( x) =
3, if x 0 2 x^2 , if 0 < x < 1 2 x, if x > 1
Evaluate the one-sided and two-sided limits of f at 1 and at 0.
§ Solution
The graph of y = f (^) ( x ) is below. It helps, but it is not required to evaluate limits. Instead, we can evaluate limits of relevant function rules.
lim x 1 ^
f (^) ( x) = lim x 1 ^
2 x^2
= 2 1( )
2
The left-hand limit as x 1 ^ : We use the rule f (^) ( x) = 2 x^2 , because it applies to a left-neighborhood of 1, say (^) (0.9, 1 ).
lim x 1 +^
f (^) ( x) = lim x 1 +^
2 x
= 2 1( )
= 2
The right-hand limit as x 1 +^ : We use the rule f (^) ( x ) = 2 x , because it applies to a right-neighborhood of 1, say (^) (1, 1.1 ).
lim x 1 f (^) ( x) = 2
The two-sided limit as x 1 : The left-hand and right-hand limits at 1 exist , and they are equal , so the two-sided limit exists and equals their common value.
lim x 0 ^
f (^) ( x) = lim x 0 ^
The left-hand limit as x 0 ^ :
We use the rule f ( x ) = 3 , because it
applies to a left-neighborhood of 0, say (^) ( 0.1, 0).
lim x 0 +^
f (^) ( x) = lim x 0 +^
2 x^2
= 2 0( )
2
The right-hand limit as x 0 +^ : We use the rule f (^) ( x ) = 2 x^2 , because it applies to a right-neighborhood of 0, say (^) (0, 0.1 ).
lim x 0 f (^) ( x )
does not exist (DNE)
The two-sided limit as x 0 : The left-hand and right-hand limits at 0 exist , but they are unequal , so the two-sided limit does not exist (DNE).
Example 12 (Infinite and/or Nonexistent Limits)
Let f (^) ( x) =
x
. Evaluate lim x 0 +^
f (^) ( x), lim x 0 ^
f (^) ( x), and lim x 0 f (^) ( x ).
§ Solution
The graph of y = f ( x ) is below. We will discuss this graph in later sections.
As x approaches 0 from the right , the function values increase without bound. Therefore, lim x 0 +^
f (^) ( x) = .
As x approaches 0 from the left , the function values decrease without bound. Therefore, lim x 0 ^
f (^) ( x) = .
and are mismatched.
Therefore, lim x 0
f (^) ( x ) does not exist (DNE).
In fact, all three limits do not exist. For example, lim x 0 +^
f (^) ( x), does not
exist , because the function values do not approach a single real number as x approaches 0 from the right. The expressions and indicate why the one-sided limits do not exist, and we write and where appropriate. §
Example 13 (Infinite and Nonexistent Limits)
Let f (^) ( x) =
x^2
. Evaluate lim x 0 +^
f (^) ( x), lim x 0 ^
f (^) ( x), and lim x 0 f (^) ( x).
§ Solution
The graph of y = f ( x ) is below. Observe that f is an even function.
lim x 0 +^
f (^) ( x) = ,
lim x 0 ^
f (^) ( x) = , and
lim x 0 f (^) ( x) = . §
Example 14 (A Nonexistent Limit)
Let f (^) ( x) =
x x
. Evaluate lim x 0 +^
f (^) ( x), lim x 0 ^
f (^) ( x), and lim x 0 f (^) ( x ).
§ Solution
Note: f is not a rational function, but it is an algebraic function , since
f (^) ( x ) =
x x
x^2 x
Remember that: x =
x , if x 0 x , if x < 0
Then, f (^) ( x) =
x x
x x
= 1, if x > 0
x x
= 1, if x < 0
, and f (^) ( ) 0 is undefined.
The graph of y = f ( x ) is below.
lim x 0 +^
f (^) ( x) = 1 ,
lim x 0 ^
f (^) ( x) = 1 , and
lim x 0
f (^) ( x) does not exist (DNE),
due to the fact that the right-hand and left-hand limits are unequal. §
