1. Home
  2. Precalculus
  3. Introduction to calculus
  4. Finding limits: numerical and

12.1 Finding limits: numerical and graphical approaches  (Page 4/10)

<< Chapter < Page
  Precalculus     Page 4 / 10
Chapter >> Page >

Using the graph of the function   y = f ( x )   shown in [link] , estimate the following limits.

Graph of a piecewise function that has three segments: 1) negative infinity to 0, 2) 0 to 2, and 3) 2 to positive inifnity, which has a discontinuity at (4, 4)

a. 0; b. 2; c. does not exist; d. − 2 ;   e. 0; f. does not exist; g. 4; h. 4; i. 4

Got questions? Get instant answers now!

Finding a limit using a table

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of   x   approach   a   from both sides. Then we determine if the output values get closer and closer to some real value, the limit       L .

Let’s consider an example using the following function:

lim x →   5 ( x 3 − 125 x − 5 )

To create the table, we evaluate the function at values close to   x = 5.   We use some input values less than 5 and some values greater than 5 as in [link] . The table values show that when   x > 5   but nearing 5, the corresponding output gets close to 75. When   x > 5   but nearing 5, the corresponding output also gets close to 75.

Table shows that as x values approach 5 from the positive or negative direction, f(x) gets very close to 75. But when x is equal to 5, y is undefined.

Because

lim x → 5 − f ( x ) = 75 = lim x → 5 + f ( x ) ,

then

lim x → 5 f ( x ) = 75.

Remember that   f ( 5 )   does not exist.

Given a function   f , use a table to find the limit as   x   approaches   a   and the value of   f ( a ) , if it exists.

  1. Choose several input values that approach   a   from both the left and right. Record them in a table.
  2. Evaluate the function at each input value. Record them in the table.
  3. Determine if the table values indicate a left-hand limit and a right-hand limit.
  4. If the left-hand and right-hand limits exist and are equal, there is a two-sided limit.
  5. Replace   x   with   a   to find the value of   f ( a ) .

Finding a limit using a table

Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.

lim x → 0 ( 5 sin ( x ) 3 x )

We can estimate the value of a limit, if it exists, by evaluating the function at values near   x = 0.   We cannot find a function value for   x = 0   directly because the result would have a denominator equal to 0, and thus would be undefined.

f ( x ) = 5 sin ( x ) 3 x

We create [link] by choosing several input values close to   x = 0 , with half of them less than   x = 0   and half of them greater than   x = 0.   Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.

The table values indicate that when   x < 0   but approaching 0, the corresponding output nears   5 3 .

When   x > 0   but approaching 0, the corresponding output also nears   5 3 .

Table shows that as x values approach 0 from the positive or negative direction, f(x) gets very close to 5 over 3. But when x is equal to 0, y is undefined.

Because

lim x → 0 − f ( x ) = 5 3 = lim x → 0 + f ( x ) ,

then

lim x → 0 f ( x ) = 5 3 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Is it possible to check our answer using a graphing utility?

Yes. We previously used a table to find a limit of 75 for the function   f ( x ) = x 3 − 125 x − 5   as   x   approaches 5. To check, we graph the function on a viewing window as shown in [link] . A graphical check shows both branches of the graph of the function get close to the output 75 as   x   nears 5. Furthermore, we can use the ‘trace’ feature of a graphing calculator. By appraoching   x = 5   we may numerically observe the corresponding outputs getting close to   75.

Graph of an increasing function with a discontinuity at (5, 75)

Numerically estimate the limit of the following function by making a table:

lim x → 0 ( 20 sin ( x ) 4 x )

lim x → 0 ( 20 sin ( x ) 4 x ) = 5

Table showing that f(x) approaches 5 from either side as x approaches 0 from either side.
Got questions? Get instant answers now!

Is one method for determining a limit better than the other?

No. Both methods have advantages. Graphing allows for quick inspection. Tables can be used when graphical utilities aren’t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph.

Questions & Answers

what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
What are the treatment for autism?
Magret Reply
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities. before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue.. sometimes due to these physical problems..the diagnosis may be misdiagnosed. treatment for autism. well it depends on the severity. since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision. play therapy. speech therapy. Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game. string and beads game. the more you interact with the child the more effective
Jharna
results you'll get.. please consult a therapist to know what suits best on your child. and last as a parent. I know sometimes it's overwhelming to guide a special kid. but trust the process and be strong and patient as a parent.
Jharna
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 4

Read also:

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