12.2 Finding limits: properties of limits

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In this section, you will:

Find the limit of a sum, a difference, and a product.
Find the limit of a polynomial.
Find the limit of a power or a root.
Find the limit of a quotient.

Consider the rational function

f (x) = \frac{x^{2} - 6 x - 7}{x - 7}

The function can be factored as follows:

f (x) = \frac{(x - 7) (x + 1)}{x - 7}, which gives us f (x) = x + 1, x \neq 7.

Does this mean the function $f$ is the same as the function $g (x) = x + 1 ?$

The answer is no. Function $f$ does not have $x = 7$ in its domain, but $g$ does. Graphically, we observe there is a hole in the graph of $f (x)$ at $x = 7,$ as shown in [link] and no such hole in the graph of $g (x),$ as shown in [link] .

Graph of an increasing function where f(x) = (x^2-6x-7)\(x-7) with a discontinuity at (7, 8) — The graph of function $f$ contains a break at *$x = 7$* and is therefore not continuous at $x = 7.$

Graph of an increasing function where g(x) = x+1 — The graph of function $g$ is continuous.

So, do these two different functions also have different limits as $x$ approaches 7?

Not necessarily. Remember, in determining a limit of a function as $x$ approaches $a,$ what matters is whether the output approaches a real number as we get close to $x = a .$ The existence of a limit does not depend on what happens when $x$ equals $a .$

Look again at [link] and [link] . Notice that in both graphs, as $x$ approaches 7, the output values approach 8. This means

\lim_{x \to 7} f (x) = \lim_{x \to 7} g (x) .

Remember that when determining a limit, the concern is what occurs near $x = a,$ not at $x = a .$ In this section, we will use a variety of methods, such as rewriting functions by factoring, to evaluate the limit. These methods will give us formal verification for what we formerly accomplished by intuition.

Finding the limit of a sum, a difference, and a product

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits , which is a collection of theorems for finding limits.

Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

A General Note

Properties of limits

Let $a, k, A,$ and $B$ represent real numbers, and $f$ and $g$ be functions, such that $\lim_{x \to a} f (x) = A$ and $\lim_{x \to a} g (x) = B .$ For limits that exist and are finite, the properties of limits are summarized in [link]

Constant, k	$\lim_{x \to a} k = k$
Constant times a function	$\lim_{x \to a} [k \cdot f (x)] = k \lim_{x \to a} f (x) = k A$
Sum of functions	$\lim_{x \to a} [f (x) + g (x)] = \lim_{x \to a} f (x) + \lim_{x \to a} g (x) = A + B$
Difference of functions	$\lim_{x \to a} [f (x) - g (x)] = \lim_{x \to a} f (x) - \lim_{x \to a} g (x) = A - B$
Product of functions	$\lim_{x \to a} [f (x) \cdot g (x)] = \lim_{x \to a} f (x) \cdot \lim_{x \to a} g (x) = A \cdot B$
Quotient of functions	$\lim_{x \to a} \frac{f (x)}{g (x)} = \frac{\lim_{x \to a} f (x)}{\lim_{x \to a} g (x)} = \frac{A}{B}, B \neq 0$
Function raised to an exponent	$\lim_{x \to a} {[f (x)]}^{n} = {[\lim_{x \to \infty} f (x)]}^{n} = A^{n},$ where $n$ is a positive integer
n th root of a function, where n is a positive integer	$\lim_{x \to a} \sqrt[n]{f (x)} = \sqrt[n]{\lim_{x \to a} [f (x)]} = \sqrt[n]{A}$
Polynomial function	$\lim_{x \to a} p (x) = p (a)$

Questions & Answers

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

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information

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devaluation

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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,

Ezea

What is ceteris paribus?

Shukri Reply

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)

Kelo

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Shukri

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what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

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

Shukri

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what is the difference between economic growth and development

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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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Jabir

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

Abdisa Reply

any question about economics?

Awais Reply

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

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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

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 .

Feyisa Reply

Answer

Feyisa

Jabir

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

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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