Proving a statement about the limit of a specific function (algebraic approach)
Prove that
Let’s use our outline from the Problem-Solving Strategy:
Let
Choose
This choice of
may appear odd at first glance, but it was obtained by taking a look at our ultimate desired inequality:
This inequality is equivalent to
At this point, the temptation simply to choose
is very strong. Unfortunately, our choice of
must depend on
ε only and no other variable. If we can replace
by a numerical value, our problem can be resolved. This is the place where assuming
comes into play. The choice of
here is arbitrary. We could have just as easily used any other positive number. In some proofs, greater care in this choice may be necessary. Now, since
and
we are able to show that
Consequently,
At this point we realize that we also need
Thus, we choose
Assume
Thus,
Since
we may conclude that
Thus, by subtracting 4 from all parts of the inequality, we obtain
Consequently,
This gives us
You will find that, in general, the more complex a function, the more likely it is that the algebraic approach is the easiest to apply. The algebraic approach is also more useful in proving statements about limits.
Proving limit laws
We now demonstrate how to use the epsilon-delta definition of a limit to construct a rigorous proof of one of the limit laws. The
triangle inequality is used at a key point of the proof, so we first review this key property of absolute value.
Definition
The
triangle inequality states that if
a and
b are any real numbers, then
Proof
We prove the following limit law: If
and
then
Let
Choose
so that if
then
Choose
so that if
then
Choose
Assume
Thus,
Hence,
□
We now explore what it means for a limit not to exist. The limit
does not exist if there is no real number
L for which
Thus, for all real numbers
L ,
To understand what this means, we look at each part of the definition of
together with its opposite. A translation of the definition is given in
[link] .
Translation of the definition of
And its opposite
Definition
Opposite
1. For every
1. There exists
so that
2. there exists a
so that
2. for every
3. if
then
3. There is an
x satisfying
so that
Finally, we may state what it means for a limit not to exist. The limit
does not exist if for every real number
L , there exists a real number
so that for all
there is an
x satisfying
so that
Let’s apply this in
[link] to show that a limit does not exist.
Questions & Answers
What are the factors that affect demand for a commodity
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
Abdureman
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