<< Chapter < Page Chapter >> Page >

Using a graphing utility to determine a limit

With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as x approaches 0. If the function has a limit as x approaches 0, state it. If not, discuss why there is no limit.

f ( x ) = 3 sin ( π x )

We can use a graphing utility to investigate the behavior of the graph close to x = 0. Centering around x = 0 , we choose two viewing windows such that the second one is zoomed in closer to x = 0 than the first one. The result would resemble [link] for [ 2 , 2 ] by [ 3 , 3 ] .

Graph of a sinusodial function zoomed in at [-2, 2] by [-3, 3].

The result would resemble [link] for [ −0.1 , 0.1 ] by [ −3 , 3 ] .

Graph of the same sinusodial function as in the previous image zoomed in at [-0.1, 0.1] by [-3. 3].
Even closer to zero, we are even less able to distinguish any limits.

The closer we get to 0, the greater the swings in the output values are. That is not the behavior of a function with either a left-hand limit or a right-hand limit. And if there is no left-hand limit or right-hand limit, there certainly is no limit to the function f ( x ) as x approaches 0.

We write

lim x 0 ( 3 sin ( π x ) )  does not exist .
lim x 0 + ( 3 sin ( π x ) )  does not exist .
lim x 0 ( 3 sin ( π x ) )  does not exist .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Numerically estimate the following limit: lim x 0 ( sin ( 2 x ) ) .

does not exist

Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with finding limits.

Key concepts

  • A function has a limit if the output values approach some value L as the input values approach some quantity a . See [link] .
  • A shorthand notation is used to describe the limit of a function according to the form lim x a f ( x ) = L , which indicates that as x approaches a , both from the left of x = a and the right of x = a , the output value gets close to L .
  • A function has a left-hand limit if f ( x ) approaches L as x approaches a where x < a . A function has a right-hand limit if f ( x ) approaches L as x approaches a where x > a .
  • A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
  • A graph provides a visual method of determining the limit of a function.
  • If the function has a limit as x approaches a , the branches of the graph will approach the same y - coordinate near x = a from the left and the right. See [link] .
  • A table can be used to determine if a function has a limit. The table should show input values that approach a from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See [link] .
  • A graphing utility can also be used to find a limit. See [link] .

Section exercises

Verbal

Explain the difference between a value at x = a and the limit as x approaches a .

The value of the function, the output, at x = a is f ( a ) . When the lim x a f ( x ) is taken, the values of x get infinitely close to a but never equal a . As the values of x approach a from the left and right, the limit is the value that the function is approaching.

Got questions? Get instant answers now!

Explain why we say a function does not have a limit as x approaches a if, as x approaches a , the left-hand limit is not equal to the right-hand limit.

Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask