4.6 Limits at infinity and asymptotes (Page 3/14)

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Definition

(Informal) We say a function $f$ has an infinite limit at infinity and write

lim_{x \to \infty} f (x) = \infty .

if $f (x)$ becomes arbitrarily large for $x$ sufficiently large. We say a function has a negative infinite limit at infinity and write

lim_{x \to \infty} f (x) = − \infty .

if $f (x) < 0$ and $| f (x) |$ becomes arbitrarily large for $x$ sufficiently large. Similarly, we can define infinite limits as $x \to − \infty .$

Formal definitions

Earlier, we used the terms arbitrarily close , arbitrarily large , and sufficiently large to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.

Definition

(Formal) We say a function $f$ has a limit at infinity , if there exists a real number $L$ such that for all $ε > 0,$ there exists $N > 0$ such that

| f (x) - L | < ε

for all $x > N .$ In that case, we write

lim_{x \to \infty} f (x) = L

(see [link] ).

We say a function $f$ has a limit at negative infinity if there exists a real number $L$ such that for all $ε > 0,$ there exists $N < 0$ such that

| f (x) - L | < ε

for all $x < N .$ In that case, we write

lim_{x \to − \infty} f (x) = L .

The function f(x) is graphed, and it has a horizontal asymptote at L. L is marked on the y axis, as is L + ॉ and L – ॉ. On the x axis, N is marked as the value of x such that f(x) = L + ॉ. — For a function with a limit at infinity, for all $x > N,$ $| f (x) - L | < ε .$

Earlier in this section, we used graphical evidence in [link] and numerical evidence in [link] to conclude that $lim_{x \to \infty} (\frac{2 + 1}{x}) = 2 .$ Here we use the formal definition of limit at infinity to prove this result rigorously.

Use the formal definition of limit at infinity to prove that $lim_{x \to \infty} (\frac{2 + 1}{x}) = 2 .$

Let $ε > 0 .$ Let $N = \frac{1}{ε} .$ Therefore, for all $x > N,$ we have

| 2 + \frac{1}{x} - 2 | = | \frac{1}{x} | = \frac{1}{x} < \frac{1}{N} = ε

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Use the formal definition of limit at infinity to prove that $lim_{x \to \infty} (\frac{3 - 1}{x^{2}}) = 3 .$

Let $ε > 0 .$ Let $N = \frac{1}{\sqrt{ε}} .$ Therefore, for all $x > N,$ we have

$| 3 - \frac{1}{x^{2}} - 3 | = \frac{1}{x^{2}} < \frac{1}{N^{2}} = ε$

Therefore, $lim_{x \to \infty} (3 - 1 / x^{2}) = 3 .$

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We now turn our attention to a more precise definition for an infinite limit at infinity.

Definition

(Formal) We say a function $f$ has an infinite limit at infinity and write

lim_{x \to \infty} f (x) = \infty

if for all $M > 0,$ there exists an $N > 0$ such that

f (x) > M

for all $x > N$ (see [link] ).

We say a function has a negative infinite limit at infinity and write

lim_{x \to \infty} f (x) = − \infty

if for all $M < 0,$ there exists an $N > 0$ such that

f (x) < M

for all $x > N .$

Similarly we can define limits as $x \to − \infty .$

The function f(x) is graphed. It continues to increase rapidly after x = N, and f(N) = M. — For a function with an infinite limit at infinity, for all $x > N,$ $f (x) > M .$

Earlier, we used graphical evidence ( [link] ) and numerical evidence ( [link] ) to conclude that $lim_{x \to \infty} x^{3} = \infty .$ Here we use the formal definition of infinite limit at infinity to prove that result.

Use the formal definition of infinite limit at infinity to prove that $lim_{x \to \infty} x^{3} = \infty .$

Let $M > 0 .$ Let $N = \sqrt[3]{M} .$ Then, for all $x > N,$ we have

x^{3} > N^{3} = {(\sqrt[3]{M})}^{3} = M .

Therefore, $lim_{x \to \infty} x^{3} = \infty .$

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Use the formal definition of infinite limit at infinity to prove that $lim_{x \to \infty} 3 x^{2} = \infty .$

Let $M > 0 .$ Let $N = \sqrt{\frac{M}{3}} .$ Then, for all $x > N,$ we have

$3 x^{2} > 3 N^{2} = 3 {(\sqrt{\frac{M}{3}})}^{2} 2 = \frac{3 M}{3} = M$

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End behavior

The behavior of a function as $x \to ± \infty$ is called the function’s end behavior . At each of the function’s ends, the function could exhibit one of the following types of behavior:

The function $f (x)$ approaches a horizontal asymptote $y = L .$
The function $f (x) \to \infty$ or $f (x) \to − \infty .$
The function does not approach a finite limit, nor does it approach $\infty$ or $− \infty .$ In this case, the function may have some oscillatory behavior.

Questions & Answers

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appreciation

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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,

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other things being equal

AI-Robot

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)

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What is different between quantity demand and demand?

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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

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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.

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What do you think is more important to focus on when considering inequality ?

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sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

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it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

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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

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 .

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Answer

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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

Gsbwnw Reply

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

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types of unemployment

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What is the difference between perfect competition and monopolistic competition?

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Source: OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2

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