Using correct notation, describe the limit of a function.
Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
Use a graph to estimate the limit of a function or to identify when the limit does not exist.
Define one-sided limits and provide examples.
Explain the relationship between one-sided and two-sided limits.
Using correct notation, describe an infinite limit.
Define a vertical asymptote.
The concept of a limit or limiting process, essential to the understanding of calculus, has been around for thousands of years. In fact, early mathematicians used a limiting process to obtain better and better approximations of areas of circles. Yet, the formal definition of a limit—as we know and understand it today—did not appear until the late 19th century. We therefore begin our quest to understand limits, as our mathematical ancestors did, by using an intuitive approach. At the end of this chapter, armed with a conceptual understanding of limits, we examine the formal definition of a limit.
We begin our exploration of limits by taking a look at the graphs of the functions
which are shown in
[link] . In particular, let’s focus our attention on the behavior of each graph at and around
Each of the three functions is undefined at
but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of
To express the behavior of each graph in the vicinity of 2 more completely, we need to introduce the concept of a limit.
Intuitive definition of a limit
Let’s first take a closer look at how the function
behaves around
in
[link] . As the values of
x approach 2 from either side of 2, the values of
approach 4. Mathematically, we say that the limit of
as
x approaches 2 is 4. Symbolically, we express this limit as
From this very brief informal look at one limit, let’s start to develop an
intuitive definition of the limit . We can think of the limit of a function at a number
a as being the one real number
L that the functional values approach as the
x -values approach
a, provided such a real number
L exists. Stated more carefully, we have the following definition:
Definition
Let
be a function defined at all values in an open interval containing
a , with the possible exception of
a itself, and let
L be a real number. If
all values of the function
approach the real number
L as the values of
approach the number
a , then we say that the limit of
as
x approaches
a is
L . (More succinct, as
x gets closer to
a ,
gets closer and stays close to
L .) Symbolically, we express this idea as
We can estimate limits by constructing tables of functional values and by looking at their graphs. This process is described in the following Problem-Solving Strategy.
Problem-solving strategy: evaluating a limit using a table of functional values
To evaluate
we begin by completing a table of functional values. We should choose two sets of
x -values—one set of values approaching
a and less than
a , and another set of values approaching
a and greater than
a .
[link] demonstrates what your tables might look like.
Table of functional values for
x
x
Use additional values as necessary.
Use additional values as necessary.
Next, let’s look at the values in each of the
columns and determine whether the values seem to be approaching a single value as we move down each column. In our columns, we look at the sequence
and so on, and
and so on. (
Note : Although we have chosen the
x -values
and so forth, and these values will probably work nearly every time, on very rare occasions we may need to modify our choices.)
If both columns approach a common
y -value
L , we state
We can use the following strategy to confirm the result obtained from the table or as an alternative method for estimating a limit.
Using a graphing calculator or computer software that allows us graph functions, we can plot the function
making sure the functional values of
for
x -values near
a are in our window. We can use the trace feature to move along the graph of the function and watch the
y -value readout as the
x -values approach
a . If the
y -values approach
L as our
x -values approach
a from both directions, then
We may need to zoom in on our graph and repeat this process several times.
Questions & Answers
differentiate between demand and supply
giving examples
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product