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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 ) = x 2 6 x 7 x 7

The function can be factored as follows:

f ( x ) = ( x 7 ) ( x + 1 ) x 7 , which gives us f ( x ) = x + 1 , x 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 7 f ( x ) = lim x 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.

Properties of limits

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

Constant, k lim x a k = k
Constant times a function lim x a [ k f ( x ) ] = k lim x a f ( x ) = k A
Sum of functions lim x a [ f ( x ) + g ( x ) ] = lim x a f ( x ) + lim x a g ( x ) = A + B
Difference of functions lim x a [ f ( x ) g ( x ) ] = lim x a f ( x ) lim x a g ( x ) = A B
Product of functions lim x a [ f ( x ) g ( x ) ] = lim x a f ( x ) lim x a g ( x ) = A B
Quotient of functions lim x a f ( x ) g ( x ) = lim x a f ( x ) lim x a g ( x ) = A B , B 0
Function raised to an exponent lim x a [ f ( x ) ] n = [ lim x 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 a f ( x ) n = lim x a [ f ( x ) ] n = A n
Polynomial function lim x a p ( x ) = p ( a )

Questions & Answers

can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
Practice Key Terms 1

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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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