MM-Limits and Continuity

(Chapter 2: Limits and Continuity) 2.0.1

CHAPTER 2: LIMITS AND CONTINUITY

In Swokowski

(Classic / 5th

ed.)

In Thomas

(11th

ed.)

2.1: An Introduction to Limits 2.1, 2.3 2.1, 2.4

2.2: Properties of Limits 2.3 2.2

2.3: Limits and Infinity I 2.4 2.4

2.4: Limits and Infinity II 2.4 2.5

2.5: The Indeterminate Forms 0/0 and / 2.1, 2.3, 2.4 2.1, 2.2, 2.4, 2.5

2.6: The Squeeze (Sandwich) Theorem 2.3 2.2

2.7: Precise Definitions of Limits 2.2 2.3

2.8: Continuity 2.5 2.6

ASSUMPTIONS THROUGHOUT THE NOTES

Unless otherwise specified …

• We assume that f and g denote functions.

• We assume that a, b, c, and k denote real constants.

• We assume that the domain of a function is its implied domain.

• We assume that graphs extend beyond the scope of the figures in an expected

manner, unless endpoints are clearly shown. Arrowheads may help to make this

clearer.

• Before we get to multivariable calculus, we will assume that “real constants” are

“real constant scalars,” as opposed to vectors.

(Section 2.1: An Introduction to Limits) 2.1.1

SECTION 2.1: AN INTRODUCTION TO LIMITS

PART A: “EASY” EXAMPLES

Example 1 (Polynomial Function)

Assuming f x( ) = 3x2+ x 1, evaluate

limx 1

f x( ) .

What is this asking?

means “approaches.” We will discuss this more rigorously later.

limx 1

f x( ) is read: “the limit of f x( ) as x approaches 1.”

It is the real number that f x( ) approaches as x approaches 1, if such a

number exists.

Solution Method

f is a polynomial function with presumably unrestricted domain R.

Here, we substitute (“plug in”) x = 1 and evaluate f 1( ) .

Warning 1: Sometimes, limx a

f x( ) does not equal f a( ) . We will see

examples of this later.

Solution

limx 1

f x( ) = limx 1

3x2+ x 1( )

Warning 2: When taking the limit of an expression

consisting of more than one term, make sure to group the

entire expression.

= 3 1( )

2

+ 1( ) 1

Warning 3: When performing substitutions, be prepared

to use grouping symbols unless you are sure that they are

not required.

= 3


Notation

We can write: limx 1

f x( ) = 3 .

Alternatively, we can write: f x( ) 3 as x 1.

A graph can demonstrate this.

Consider the graph of y = f x( ) ; here, y = 3x2+ x 1.

(Figure 2.1.a)

Imagine that the arrows in the figure above represent two lovers

running towards each other along the parabola. What is the

y-coordinate of the point that they are approaching? It is 3.

(Remember that y-coordinates correspond to function values here.)

Example 2 (Rational Function)

Assuming f x( ) =2x +1

x 2, evaluate

limx 3

f x( ) .

Solution Method

f is a rational function with implied domain

x R x 2{ } .

Here, we observe that 3 is in the domain of f , so we substitute

(“plug in”) x = 3 and evaluate f 3( ) .


Solution

limx 3

f x( ) = limx 3

2x +1

x 2

=2 3( ) +1

3( ) 2

= 7


Consider the graph of y = f x( ) ; here, y =

2x +1

x 2.

(Figure 2.1.b)

Note: You might not know how to graph the entire graph until later.

This will often be the case with these figures.

But wait! What if the lover on the left is running along the left branch of the

graph?

The left branch is irrelevant to our analysis. We really only care about

what happens when the lovers are in the “immediate vicinity” of

x = 3; this will be explained rigorously later.

LIMITS ARE “LOCAL.”

By the way, what are the brown dashed lines?

They’re called vertical and horizontal asymptotes (VAs and HAs),

which are lines that the graph approaches in a “long-run” or

“explosive” sense. We’ll define them using limits later on.


Example 3 (Constant Function)

lim

x

2 = 2 .


Think: f x( ) = 2 . Consider the graph of

y = 2 .

(Figure 2.1.c)

The term “approaches”

We can write 2 2 (i.e, 2 approaches 2) as x .

This is true, even though people are more used to thinking about a

sequence of distinct numbers such as 2.1, 2.01, 2.001, … approaching

2. It helps to remember that, in math, the constant sequence 2, 2, 2, …

is also said to approach 2.

Remember that all constant functions are also polynomial functions, and all

polynomial functions are also rational functions.

A Limit Theorem for Rational Functions

If:

f is a rational function;

its domain, Dom f( ) , is its implied domain; and

a is a real constant in Dom f( ) , then:

limx a

f x( ) = f a( ) .

That is, to compute the limit, substitute (“plug in”) x = a and

evaluate f a( ) .

We will justify this theorem later.


Be prepared to work with function and variable names other than f and x.

Example 4 (Revisiting Example 2)

Assuming g t( ) =

2t +1

t 2, evaluate

limt 3

g t( ) .

Solution

limt 3

g t( ) = limt 3

2t +1

t 2

=2 3( ) +1

3( ) 2

= 7

PART B: ONE- AND TWO-SIDED LIMITS; EXISTENCE OF LIMITS

The limit problems of the form limx a

f x( ) in Part A were two-sided limit problems,

because we were interested in what happened when we approached x = a from

both sides: from the left and from the right.

One-sided limit problems only focus on one of these approaches.

Example 5 (Left-Hand Limit)

Evaluate limx 3

x + 3( ) .


This is an example of a left-hand limit problem, which is a type of

one-sided limit problem.

Let f x( ) = x + 3 . lim

x 3

f x( ) is read:

“the limit of f x( ) as x approaches 3 from the left.”

It is the real number that f x( ) approaches as x approaches 3 from

lesser or lower numbers (imagine approaching x = 3 from the left

along the real number line), if such a number exists.


Solution Method

We use the same method that we used in Part A for two-sided limits.

f is a rational (in fact, polynomial) function with implied domain R.

Here, we observe that 3 is in the domain of f, so we substitute

(“plug in”) x = 3 and evaluate f 3( ) .

Solution

limx 3

x + 3( ) = 3+ 3

= 6

Numerical / Tabular Method

Although it is not a comprehensively convincing method that is

typically accepted on exams, and it can even be misleading at times,

this method is at least easy to understand.

Select an increasing sequence of real numbers that approaches 3 such

that all the numbers are less than 3. Then, evaluate the function at

those numbers and take a guess as to what number, if any, the

function values are approaching. For example:

x f x( ) = x + 3

2.9 5.9

2.99 5.99

2.999 5.999

3

6 (?)

or

x 2.9 2.99 2.999 3

f x( ) = x + 3 5.9 5.99 5.999 6 (?)

We guess:

limx 3

x + 3( ) = 6 .

Warning 4: Be careful about associating the “ ” superscript with

negative numbers. Here, when considering the left-hand limit, we

want to consider positive numbers that are close to (but less than) 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.



Consider the graph of y = f x( ) ; here, y = x + 3 .

(Figure 2.1.d)

We only care about the lover to the left of x = 3.

(A stand-up, perhaps?)

Example 6 (Right-Hand Limit)

Evaluate

limx 3

+x + 3( ) .


This is an example of a right-hand limit problem, which is a type of

one-sided limit problem.

Let f x( ) = x + 3 .

lim

x 3+

f x( ) is read:

“the limit of f x( ) as x approaches 3 from the right.”

It is the real number that f x( ) approaches as x approaches 3 from

greater or higher numbers (imagine approaching x = 3 from the right

along the real number line), if such a number exists.

Solution

lim

x 3+

x + 3( ) = 3+ 3

= 6


Numerical / Tabular Method

Select a decreasing sequence of real numbers that approaches 3 such

that all the numbers are greater than 3. Then, evaluate the function at

those numbers and take a guess as to what number, if any, the

function values are approaching. For example:

x f x( ) = x + 3

3.1 6.1

3.01 6.01

3.001 6.001

3

6 (?)

or

x 3 3.001 3.01 3.1

f x( ) = x + 3 6 (?) 6.001 6.01 6.1

We guess:

limx 3

+x + 3( ) = 6 .


Consider the graph of y = f x( ) ; here,

y = x + 3 .

(Figure 2.1.e)

We only care about the lover to the right of x = 3.

(Maybe the lovers have bad timing?)


Existence of Limits

A limit exists (if and only if, or iff) the limit can be expressed as a

single real constant. Otherwise, the limit does not exist (“DNE”).

Later, we will be able to say that a limit is (infinity) or (negative

infinity) in some cases, but the limit is still nonexistent in those cases. The

notation in those cases indicates why the limit does not exist.

Two-Sided Limits

If a and L are real constants,

then limx a

f x( ) = L

(

limx a

f x( ) = L , and

limx a+

f x( ) = L ).

That is, a two-sided limit exists the left-hand and right-hand limits

exist, and they equal the same real constant. The value of the two-sided

limit then equals that constant.

If either one-sided limit does not exist (DNE), or if the two one-sided

limits exist but are unequal, then the two-sided limit does not exist (DNE).

Example 7 (Revisiting Examples 5 and 6)

limx 3

x + 3( ) = 6 , and

limx 3

+x + 3( ) = 6 , so

limx 3

x + 3( ) = 6 .


Revisiting our Limit Theorem for Rational Functions

If:

f is a rational function;

its domain, Dom f( ) , is its implied domain; and

a is a real constant in Dom f( ) ,

then:

lim

x af x( ) = f a( ) ,

lim

x a+f x( ) = f a( ) , and, therefore,

limx a

f x( ) = f a( ) .

That is, to compute these limits, substitute (“plug in”) x = a and

evaluate f a( ) .

Let’s look at a function that is not rational. The aforementioned theorem will not

apply.

Example 8

Let f x( ) = x +1 .

Observe that Dom f( ) = x R x 0{ } , or 0, ) .

(Figure 2.1.f)


Right-Hand Limit: lim

x 0+

f x( ) = 1.

This is because the function values approach 1 as we approach

x = 0 from the right.

Left-Hand Limit:

limx 0

f x( ) does not exist (DNE).

This is because function values in this example are undefined as

we approach x = 0 from the left. In order for the left-hand limit

to exist, there must exist an interval of the form c, 0( ) on

(i.e., throughout) which the function is defined, for some

negative real constant c. That is not the case here.

By the way, why did we give the interval form c, 0( ) and not

c, 0( above? We do not require the function to be defined at 0,

itself! We will discuss this issue later.

Two-Sided Limit: limx 0


This is because the corresponding left-hand limit does not exist

(DNE).

Observe that f is not a rational function, so the aforementioned (revised)

Limit Theorem for Rational Functions does not apply, even though 0 is in

Dom f( ) . f is, however, an algebraic function, and we will discuss a limit

theorem for algebraic functions later.


PART C: THERE DOESN’T HAVE TO BE A “POINT”!

“IGNORE a” THEOREMS

Example 9 (Modifying Examples 5-7)

Let g x( ) = x + 3, x 3( ) . We are removing 3 from the domain of the function from Examples 5-7.

(Figure 2.1.g)

The point 3, 6( ) is no longer on the graph. Instead, we have a hole;

later, we will say that there is a removable discontinuity at x = 3.

Nevertheless, even though f 3( ) is now undefined, the following

statements are true:

lim

x 3

g x( ) = 6 ,

lim

x 3+

g x( ) = 6 , and

limx 3

g x( ) = 6 .

In Examples 5-7, the limit value was attained by the function at x = 3.

Here, it is not!


f a( ) may or may not be relevant to limx a

f x( )

The existence of limx a

f x( ) does not require the existence of f a( ) .

(See Example 9.)

Even if f a( ) exists,

limx a

f x( ) could be a different value, or it might not

exist at all. (See Example 10.)

If limx a

f x( ) = f a( ) , then f is continuous at a, as it was in

Examples 5-7 for a = 3; we will discuss continuity later.

Example 10 (Modifying Example 9)

Let the function h be defined piecewise as follows:

h x( ) =x + 3, x 3

7, x = 3

(A piecewise-defined function uses different evaluation rules for different

subsets of – i.e., different groups of values in – its domain. This type of

function can lead to interesting limit problems.)

Evaluate limx 3

h x( ) .

What does the graph of h suggest?

(Figure 2.1.h)

limx 3

h x( ) = 6 once again, even though h 3( ) = 7 .


Suggested solution

limx 3

h x( ) = limx 3

x + 3( )

= 3+ 3

= 6

Why is the suggested solution appropriate?

We only care about the behavior of the h function in the “immediate

vicinity” of x = 3, excluding x = 3, itself.

The function rule h x( ) = x + 3 applies to the values of x that are in the

“immediate vicinity” of x = 3, excluding x = 3, itself.

More precisely, we can find an open interval containing 3, say

2.9, 3.1( ) or even the entirety of R, on which the h function is defined

using the function rule h x( ) = x + 3, except at x = 3, itself.

Therefore, h x( ) = x + 3 is the only rule that is relevant when we

consider approaching x = 3 from the left or from the right.

As a consequence, either limx 3

h x( ) = limx 3

x + 3( ) , or neither limit

exists. We know limx 3

x + 3( ) = 6 , so we can conclude that

limx 3

h x( ) = 6.


How do we generalize this approach? (Perhaps look at Example 11 now.)

The "Ignore a" Theorem for Two-Sided Limits:

Evaluating the two-sided limit limx a

f x( )

even if f is not a rational function with a in its domain

If:

f is a function that is defined by the function rule

r x( ) on (i.e., throughout) some open x-interval

containing the real constant a, possibly excluding a,

itself,

then:

limx a

f x( ) = limx a

r x( ) , or neither limit exists.

We can develop modified theorems for one-sided limits as follows.

These modifications will be made clearer in Example 11.

Basically, when evaluating a left-hand limit, we use the function rule

that governs the x-values “immediately to the left” of a on the real

number line. Likewise, when evaluating a right-hand limit, we use the

rule that governs the x-values “immediately to the right” of a.

The "Ignore a" Theorem for Left-Hand Limits

If:


r x( ) on (i.e., throughout) some open x-interval of

the form c, a( ) , where c is a real constant and c < a ,

then:

lim

x af x( ) = lim

x ar x( ) , or neither limit exists.

The "Ignore a" Theorem for Right-Hand Limits

If:


r x( ) on (i.e., throughout) some open x-interval of

the form a, c( ) , where c is a real constant and c > a ,

then:

limx a+

f x( ) = limx a+

r x( ) , or neither limit exists.


What’s the bottom line? Does it matter what happens to a function f at a

or not when we evaluate limits like lim

x af x( ) ?

In theory, we’re not supposed to care what happens to the function

at x = a .

Often, though, it helps to know what happens at x = a .

For example, if we have a rational function f that has a in its

domain (and the domain is the implied domain), then

limx a

f x( ) = f a( ) . (We will come back to this issue when we cover

continuity.)

Example 11

Let the function f be defined piecewise as follows:

f x( ) =3, if x 0

2x2, if 0 < x < 1

2x, if x > 1

Consider the graph of y = f x( ) .

(Figure 2.1.i)


limx 1

f x( ) = limx 1

2x2

= 2 1( )2

= 2

The left-hand limit as x 1 :

The relevant function rule is 2x2, because

that rule applies to the x-values in an open

interval of the form c, 1( ) , where c < 1;

for example, consider the interval 0.9,1( ) .

limx 1

+f x( ) = lim

x 1+

2x

= 2 1( )= 2

The right-hand limit as x 1+ :

The relevant function rule is 2x , because


interval of the form 1, c( ) , where c > 1;

for example, consider the interval 1,1.1( ) .

limx 1

f x( ) = 2

The two-sided limit as x 1:

The left-hand and right-hand limits as

x 1 exist and are equal, so the two-sided

limit exists and equals their common value.

limx 0

f x( ) = limx 0

3

= 3

The left-hand limit as x 0 :

The relevant function rule is 3, because


interval of the form c, 0( ) , where c < 0 ; for

example, consider the interval 0.1, 0( ) .

limx 0

+f x( ) = lim

x 0+

2x2

= 2 0( )2

= 0

The right-hand limit as x 0 + :

The relevant function rule is 2x2, because


interval of the form 0, c( ) , where c > 0 ;

for example, consider the interval 0, 0.1( ) .

limx 0

f x( )

does not exist (DNE)

The two-sided limit as x 0 :

The left-hand and right-hand limits as

x 0 exist but are unequal, so the

two-sided limit does not exist (DNE).


PART D: MORE EXAMPLES OF LIMITS THAT DO NOT EXIST (DNE)

Example 11

Let

f x( ) = sin1

x.

Evaluate limx 0

f x( ) , limx 0

+f x( ) , and lim

x 0f x( ) .

(Figure 2.1.j)

As x approaches 0 from the left or from the right, the function values

oscillate between

1 and 1. They do not approach a single real

constant as x approaches 0 from the left, nor from the right. Therefore,

lim

x 0

f x( ) does not exist (DNE),

lim

x 0+

f x( ) does not exist (DNE), and

limx 0


We say that we have “evaluated” limx 0

f x( ) , even though the limit

does not exist (DNE) and has no real value.

Note: The y-axis is not a vertical asymptote (VA) here, because the

graph and the function values are not “exploding” around the y-axis.


Example 12

Let f x( ) =

1

x.

(Figure 2.1.k)

When we discuss (infinity) and (negative infinity) in a later

section, we will be able to say:

lim

x 0

f x( ) = ,

lim

x 0+

f x( ) = , and

limx 0


In fact, all three indicated limits do not exist;

the first two statements indicate why those limits do not exist.

lim

x 0

f x( ) , for example, does not exist, because the function values

do not approach a single real constant as x approaches 0 from the left.

We will revisit this function in later sections.


Example 13

Let f x( ) =

x

x.

Note: f is not a rational function, but it is an algebraic function, since

f x( ) =

x

x=

x2

x.

Remember the piecewise definition of x :

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.

(Figure 2.1.letter l)

lim

x 0

f x( ) = 1,

lim

x 0+

f x( ) = 1, and

limx 0

f x( ) does not exist (DNE)

due to the fact that the aforementioned left-hand

and right-hand limits exist but are unequal.

(Section 2.1: An Introduction to Limits) 2.1.21.

FOOTNOTES

1. Limits do not require continuity. In a later section, we will discuss continuity, a property of

many functions that helps the lovers run along the graph of a function without having to jump

or hop. In the first few problems of this section, we had the luxury of imagining the lovers

running towards each other (one from the left, one from the right) while staying on the graph

of f and without having to jump or hop, provided they were placed on appropriate parts of

the graph. Sometimes, the “run” really requires jumping or hopping. For example, consider

the following function f. It turns out to be true that limx 0

f x( ) = 0 .

f x( ) =0, if x is a rational value

x, if x is an irrational value

2. Misconceptions about limits.

See “Why Is the Limit Concept So Difficult for Students?” by Sally Jacobs in the Fall 2002

edition (vol.24, No.1) of The AMATYC Review, pp.25-34.

• Students can be misled by the use of the word “limit” in real-world contexts. For example, a

speed limit is a bound that is not supposed to be exceeded; there is no such restriction on

limits in calculus.

• Limit values can sometimes be attained. For example, if a function f is continuous at

x = a (see Examples 5-7 for a = 3 ), then the function value takes on the limit value at

x = a .

• Limit values do not have to be attained. See Examples 9 and 10.

Observations:

• The dynamic view of limits, which involve ideas of motion and approaching (for example,

our lovers), may be more accessible to students than the static view preferred by many

textbook authors. The static view is exemplified by the formal definitions of limits we will

see later. The dynamic view greatly assists students in transitioning to the static view and the

formal definitions.

• Leading mathematicians in 18th- and 19

th-century Europe had heated debates about ideas of

limits.

3. Multivariable calculus. When we go to higher dimensions, there will be more than two

possible approaches when dealing with limit problems!

4. An example where a left-hand limit exists but not the right-hand limit.

Let

g x( ) =x + x 1+ x( )

xsin

1

x=

x sin1

x, if x < 0

2 + x( )sin1

x, if x > 0

. (Figure 2.1.m)

Then,

limx 0

f x( ) = 0 , which can be proven by the Squeeze (Sandwich) Theorem, something

we will cover in a later section. However,

limx 0+


See William F. Trench, Introduction to Real Analysis (free online), p.39.

MM-Limits and Continuity

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