# 2.3 The limit laws

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• Recognize the basic limit laws.
• Use the limit laws to evaluate the limit of a function.
• Evaluate the limit of a function by factoring.
• Use the limit laws to evaluate the limit of a polynomial or rational function.
• Evaluate the limit of a function by factoring or by using conjugates.
• Evaluate the limit of a function by using the squeeze theorem.

In the previous section, we evaluated limits by looking at graphs or by constructing a table of values. In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits.

## Evaluating limits with the limit laws

The first two limit laws were stated in [link] and we repeat them here. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions.

## Basic limit results

For any real number a and any constant c ,

1. $\underset{x\to a}{\text{lim}}x=a$
2. $\underset{x\to a}{\text{lim}}c=c$

## Evaluating a basic limit

Evaluate each of the following limits using [link] .

1. $\underset{x\to 2}{\text{lim}}x$
2. $\underset{x\to 2}{\text{lim}}5$
1. The limit of x as x approaches a is a : $\underset{x\to 2}{\text{lim}}x=2.$
2. The limit of a constant is that constant: $\underset{x\to 2}{\text{lim}}5=5.$

We now take a look at the limit laws    , the individual properties of limits. The proofs that these laws hold are omitted here.

## Limit laws

Let $f\left(x\right)$ and $g\left(x\right)$ be defined for all $x\ne a$ over some open interval containing a . Assume that L and M are real numbers such that $\underset{x\to a}{\text{lim}}f\left(x\right)=L$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=M.$ Let c be a constant. Then, each of the following statements holds:

Sum law for limits : $\underset{x\to a}{\text{lim}}\left(f\left(x\right)+g\left(x\right)\right)=\underset{x\to a}{\text{lim}}f\left(x\right)+\underset{x\to a}{\text{lim}}g\left(x\right)=L+M$

Difference law for limits : $\underset{x\to a}{\text{lim}}\left(f\left(x\right)-g\left(x\right)\right)=\underset{x\to a}{\text{lim}}f\left(x\right)-\underset{x\to a}{\text{lim}}g\left(x\right)=L-M$

Constant multiple law for limits : $\underset{x\to a}{\text{lim}}cf\left(x\right)=c·\underset{x\to a}{\text{lim}}f\left(x\right)=cL$

Product law for limits : $\underset{x\to a}{\text{lim}}\left(f\left(x\right)·g\left(x\right)\right)=\underset{x\to a}{\text{lim}}f\left(x\right)·\underset{x\to a}{\text{lim}}g\left(x\right)=L·M$

Quotient law for limits : $\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to a}{\text{lim}}f\left(x\right)}{\underset{x\to a}{\text{lim}}g\left(x\right)}=\frac{L}{M}$ for $M\ne 0$

Power law for limits : $\underset{x\to a}{\text{lim}}{\left(f\left(x\right)\right)}^{n}={\left(\underset{x\to a}{\text{lim}}f\left(x\right)\right)}^{n}={L}^{n}$ for every positive integer n .

Root law for limits : $\underset{x\to a}{\text{lim}}\sqrt[n]{f\left(x\right)}=\sqrt[n]{\underset{x\to a}{\text{lim}}f\left(x\right)}=\sqrt[n]{L}$ for all L if n is odd and for $L\ge 0$ if n is even.

We now practice applying these limit laws to evaluate a limit.

## Evaluating a limit using limit laws

Use the limit laws to evaluate $\underset{x\to -3}{\text{lim}}\left(4x+2\right).$

Let’s apply the limit laws one step at a time to be sure we understand how they work. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied.

$\begin{array}{ccccc}\underset{x\to -3}{\text{lim}}\left(4x+2\right)\hfill & =\underset{x\to -3}{\text{lim}}4x+\underset{x\to -3}{\text{lim}}2\hfill & & & \text{Apply the sum law.}\hfill \\ & =4·\underset{x\to -3}{\text{lim}}x+\underset{x\to -3}{\text{lim}}2\hfill & & & \text{Apply the constant multiple law.}\hfill \\ & =4·\left(-3\right)+2=-10.\hfill & & & \text{Apply the basic limit results and simplify.}\hfill \end{array}$

## Using limit laws repeatedly

Use the limit laws to evaluate $\underset{x\to 2}{\text{lim}}\frac{2{x}^{2}-3x+1}{{x}^{3}+4}.$

To find this limit, we need to apply the limit laws several times. Again, we need to keep in mind that as we rewrite the limit in terms of other limits, each new limit must exist for the limit law to be applied.

$\begin{array}{}\\ \\ \underset{x\to 2}{\text{lim}}\frac{2{x}^{2}-3x+1}{{x}^{3}+4}\hfill & =\frac{\underset{x\to 2}{\text{lim}}\left(2{x}^{2}-3x+1\right)}{\underset{x\to 2}{\text{lim}}\left({x}^{3}+4\right)}\hfill & & & \text{Apply the quotient law, making sure that.}\phantom{\rule{0.2em}{0ex}}{\left(2\right)}^{3}+4\ne 0\hfill \\ & =\frac{2·\underset{x\to 2}{\text{lim}}{x}^{2}-3·\underset{x\to 2}{\text{lim}}x+\underset{x\to 2}{\text{lim}}1}{\underset{x\to 2}{\text{lim}}{x}^{3}+\underset{x\to 2}{\text{lim}}4}\hfill & & & \text{Apply the sum law and constant multiple law.}\hfill \\ & =\frac{2·{\left(\underset{x\to 2}{\text{lim}}x\right)}^{2}-3·\underset{x\to 2}{\text{lim}}x+\underset{x\to 2}{\text{lim}}1}{{\left(\underset{x\to 2}{\text{lim}}x\right)}^{3}+\underset{x\to 2}{\text{lim}}4}\hfill & & & \text{Apply the power law.}\hfill \\ & =\frac{2\left(4\right)-3\left(2\right)+1}{{\left(2\right)}^{3}+4}=\frac{1}{4}.\hfill & & & \text{Apply the basic limit laws and simplify.}\hfill \end{array}$

Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
show that lim f(x) + lim g(x)=m+l
list the basic elementary differentials
Differentiation and integration
yes
Damien
proper definition of derivative
the maximum rate of change of one variable with respect to another variable
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
what is calculus?
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
what is x and how x=9.1 take?
what is f(x)
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
I dont understand what you wanna say by (A' n B^c)^c'
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Ok so the set is formed by vectors and not numbers
A vector of length n
But you can make a set out of matrixes as well
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
High-school?
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
I would say 24
Offer both
Sorry 20
Actually you have 40 - 4 =36 who offer maths or physics or both.
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
56-36=20 who give both courses... I would say that
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
how do i evaluate integral of x^1/2 In x
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
by using integration product formula
Roha
find derivative f(x)=1/x
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
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Roha
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do u mean fundamental theorem ?
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