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

12.1 Finding limits: numerical and graphical approaches  (Page 3/10)

<< Chapter < Page
  Precalculus     Page 3 / 10
Chapter >> Page >

Understanding two-sided limits

In the previous example, the left-hand limit and right-hand limit as   x   approaches   a   are equal. If the left- and right-hand limits are equal, we say that the function   f ( x )   has a two-sided limit    as   x   approaches   a .   More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.

The two-sided limit of function as x Approaches a

The limit of a function   f ( x ) , as   x   approaches   a , is equal to   L , that is,

lim x → a   f ( x ) = L

if and only if

lim x → a − f ( x ) = lim x → a + f ( x ) .

In other words, the left-hand limit of a function   f ( x )   as   x   approaches   a   is equal to the right-hand limit of the same function as   x   approaches   a .   If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.

Finding a limit using a graph

To visually determine if a limit exists as   x   approaches   a , we observe the graph of the function when   x   is very near to   x = a .   In [link] we observe the behavior of the graph on both sides of   a .

Graph of a function that explains the behavior of a limit at (a, L) where the function is increasing when x is less than a and decreasing when x is greater than a.

To determine if a left-hand limit exists, we observe the branch of the graph to the left of   x = a , but near   x = a .   This is where   x < a .   We see that the outputs are getting close to some real number   L   so there is a left-hand limit.

To determine if a right-hand limit exists, observe the branch of the graph to the right of   x = a , but near   x = a .   This is where   x > a .   We see that the outputs are getting close to some real number   L , so there is a right-hand limit.

If the left-hand limit and the right-hand limit are the same, as they are in [link] , then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.

Finally, we can look for an output value for the function   f ( x )   when the input value   x   is equal to   a .   The coordinate pair of the point would be   ( a , f ( a ) ) .   If such a point exists, then   f ( a )   has a value. If the point does not exist, as in [link] , then we say that   f ( a )   does not exist.

Given a function   f ( x ) , use a graph to find the limits and a function value as   x   approaches   a .

  1. Examine the graph to determine whether a left-hand limit exists.
  2. Examine the graph to determine whether a right-hand limit exists.
  3. If the two one-sided limits exist and are equal, then there is a two-sided limit—what we normally call a “limit.”
  4. If there is a point at   x = a , then   f ( a )   is the corresponding function value.

Finding a limit using a graph

  1. Determine the following limits and function value for the function   f   shown in [link] .
    1. lim x → 2 − f ( x )
    2. lim x → 2 + f ( x )
    3. lim x → 2 f ( x )
    4. f ( 2 )
    Graph of a piecewise function that has a positive parabola centered at the origin and goes from negative infinity to (2, 8), an open point, and a decreasing line from (2, 3), a closed point, to positive infinity on the x-axis.
  2. Determine the following limits and function value for the function   f   shown in [link] .
    1. lim x → 2 − f ( x )
    2. lim x → 2 + f ( x )
    3. lim x → 2 f ( x )
    4. f ( 2 )
    Graph of a piecewise function that has a positive parabola from negative infinity to 2 on the x-axis, a decreasing line from 2 to positive infinity on the x-axis, and a point at (2, 4).
  1. Looking at [link] :
    1. lim x → 2 − f ( x ) = 8 ;   when   x < 2 , but infinitesimally close to 2, the output values get close to   y = 8.
    2. lim x →   2   + f ( x ) = 3 ;   when   x > 2 , but infinitesimally close to 2, the output values approach   y = 3.
    3. lim x →   2 f ( x )   does not exist because   lim x →   2   − f ( x ) ≠ lim x →   2   + f ( x ) ;   the left and right-hand limits are not equal.
    4. f ( 2 ) = 3   because the graph of the function   f   passes through the point   ( 2 , f ( 2 ) )   or   ( 2 , 3 ) .
  2. Looking at [link] :
    1. lim x →   2   − f ( x ) = 8 ;   when   x < 2   but infinitesimally close to 2, the output values approach   y = 8.
    2. lim x →   2   + f ( x ) = 8 ;   when   x > 2   but infinitesimally close to 2, the output values approach   y = 8.
    3. lim x →   2 f ( x ) = 8   because   lim x →   2   − f ( x ) = lim x →   2   + f ( x ) = 8 ;   the left and right-hand limits are equal.
    4. f ( 2 ) = 4   because the graph of the function   f   passes through the point   ( 2 , f ( 2 ) )   or   ( 2 , 4 ) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

summarize halerambos & holbon
David Reply
the Three stages of Auguste Comte
Clementina Reply
what are agents of socialization
Antonio Reply
sociology of education
Nuhu Reply
definition of sociology of education
Nuhu
what is culture
Abdulrahim Reply
shared beliefs, values, and practices
AI-Robot
What are the two type of scientific method
ogunniran Reply
I'm willing to join you
Aceng Reply
what are the scientific method of sociology
Man
what is socialization
ogunniran Reply
the process wherein people come to understand societal norms and expectations, to accept society's beliefs, and to be aware of societal values
AI-Robot
scientific method in doing research
ogunniran
defimition of sickness in afica
Anita
Cosmology
ogunniran
Hmmm
ogunniran
list and explain the terms that found in society
REMMY Reply
list and explain the terms that found in society
Mukhtar
what are the agents of socialization
Antonio
Family Peer group Institution
Abdulwajud
I mean the definition
Antonio
ways of perceived deviance indifferent society
Naomi Reply
reasons of joining groups
SAM
to bring development to the nation at large
Hyellafiya
entails of consultative and consensus building from others
Gadama
World first Sociologist?
Abu
What is evolutionary model
Muhammad Reply
Evolution models refer to mathematical and computational representations of the processes involved in biological evolution. These models aim to simulate and understand how species change over time through mechanisms such as natural selection, genetic drift, and mutation. Evolutionary models can be u
faruk
what are the modern trends in religious behaviours
Selekeye Reply
what are social norms
Daniel Reply
shared standards of acceptable behavior by the group or appropriate behavior in a particular institution or those behaviors that are acceptable in a society
Lucius
that is how i understood it
Lucius
examples of societal norms
Diamond
Discuss the characteristics of the research located within positivist and the interpretivist paradigm
Tariro Reply
what is Industrialisation
Selekeye Reply
industrialization
Angelo
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