2.1 Limits of Functions 1
2.1: Limits of Functions
Textbook: CLP-I 1.3
Objectives
Define the limit of a function
Define one-sided and two-sided limits
Use graphs of functions to determine the values of limits or if a limit does not exist
Motivation
Before we begin discussing calculus itself, it is important to introduce limits and to
understand the logic and calculations behind them. The concept of a limit is essential to
calculus and is something that underlies the two fundamental operations of calculus:
differentiation and integration.
We will, of course, discuss these two operations and their uses in much more detail as
discussing limits.
In this set of notes, we will focus on how we can use limits to describe the behavior of a
function.
Describing the Behavior of a Function
Consider a situation where we have a quantity that may not be constant within the
context of a problem or experiment. Though the value of this quantity may not be
constant, we still might want to represent that quantity in an equation or formula. So
any value that the quantity could assume. If we use to denote that quantity, then we
call a variable.
⽐尔
Examples
cwitrridm variable
= the time (in seconds) since an object was dropped from the top of a building
= the altitude (in feet) of a plane
= the number of units of a product that are sold
Often times we wish to describe how other quantities change as the variable changes.
些
The relationship between and some other quantity that changes as changes can be
expressed as a function.
A function is a rule that associates a unique output with each input to the function. If
is the input to the function, then we denote the output of the function as . This is
© 2018-2022 Claudia Mahler
, 2.1 Limits of Functions 2
Examples
= the position of an object as a function of the time since the object was
dropped from the top of a building
= the surrounding atmospheric pressure as a function of the altitude of a
plane
= the total profit as a function of the number of units of a product that are sold
Note: if you wish to review functions and related concepts such as the domain and range, please
refer to CLP-I 0.4
回
(physics, finance, chemistry, music, etc.). However, for the purposes of introducing
many of the definitions and ideas that we will discuss in this class, we will often rely on
used to demonstrate the concepts we wish to learn.
formula
Examples
domain
xE IR the real line
XE R
not Welldefined
R except ㄨ 1 fan⼆六 ⼆ jc
回
local and long-term behavior.
The local behavior of a function describes what happens to as the value of x
approaches a specific number.
Example 2.1.1
Let . Describe the local behavior of this function as approaches 1.
fun⼆ 点
arent
ㄨ o f x ⼆千 ⼆ 1
0.9 0.5263158 ㄨ 1 fx ⼆六⼆三
1 0.999 0.5002501
舀
0.99999
0 5000025
ㄨ1 1.00001 0.4999975
⽓
1.001 0.4997501
i 1.1 0.4761905
© 2018-2022 Claudia Mahler
2.1: Limits of Functions
Textbook: CLP-I 1.3
Objectives
Define the limit of a function
Define one-sided and two-sided limits
Use graphs of functions to determine the values of limits or if a limit does not exist
Motivation
Before we begin discussing calculus itself, it is important to introduce limits and to
understand the logic and calculations behind them. The concept of a limit is essential to
calculus and is something that underlies the two fundamental operations of calculus:
differentiation and integration.
We will, of course, discuss these two operations and their uses in much more detail as
discussing limits.
In this set of notes, we will focus on how we can use limits to describe the behavior of a
function.
Describing the Behavior of a Function
Consider a situation where we have a quantity that may not be constant within the
context of a problem or experiment. Though the value of this quantity may not be
constant, we still might want to represent that quantity in an equation or formula. So
any value that the quantity could assume. If we use to denote that quantity, then we
call a variable.
⽐尔
Examples
cwitrridm variable
= the time (in seconds) since an object was dropped from the top of a building
= the altitude (in feet) of a plane
= the number of units of a product that are sold
Often times we wish to describe how other quantities change as the variable changes.
些
The relationship between and some other quantity that changes as changes can be
expressed as a function.
A function is a rule that associates a unique output with each input to the function. If
is the input to the function, then we denote the output of the function as . This is
© 2018-2022 Claudia Mahler
, 2.1 Limits of Functions 2
Examples
= the position of an object as a function of the time since the object was
dropped from the top of a building
= the surrounding atmospheric pressure as a function of the altitude of a
plane
= the total profit as a function of the number of units of a product that are sold
Note: if you wish to review functions and related concepts such as the domain and range, please
refer to CLP-I 0.4
回
(physics, finance, chemistry, music, etc.). However, for the purposes of introducing
many of the definitions and ideas that we will discuss in this class, we will often rely on
used to demonstrate the concepts we wish to learn.
formula
Examples
domain
xE IR the real line
XE R
not Welldefined
R except ㄨ 1 fan⼆六 ⼆ jc
回
local and long-term behavior.
The local behavior of a function describes what happens to as the value of x
approaches a specific number.
Example 2.1.1
Let . Describe the local behavior of this function as approaches 1.
fun⼆ 点
arent
ㄨ o f x ⼆千 ⼆ 1
0.9 0.5263158 ㄨ 1 fx ⼆六⼆三
1 0.999 0.5002501
舀
0.99999
0 5000025
ㄨ1 1.00001 0.4999975
⽓
1.001 0.4997501
i 1.1 0.4761905
© 2018-2022 Claudia Mahler
