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Evaluating the limit of a function algebraically

Evaluate lim x 3 ( 2 x + 5 ) .

lim x 3 ( 2 x + 5 ) = lim x 3 ( 2 x ) + lim x 3 ( 5 ) Sum of functions property                       = 2 lim x 3 ( x ) + lim x 3 ( 5 ) Constant times a function property                       = 2 ( 3 ) + 5   Evaluate                       = 11
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Evaluate the following limit: lim x 12 ( 2 x + 2 ) .

26

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Finding the limit of a polynomial

Not all functions or their limits involve simple addition, subtraction, or multiplication. Some may include polynomials. Recall that a polynomial is an expression consisting of the sum of two or more terms, each of which consists of a constant and a variable raised to a nonnegative integral power. To find the limit of a polynomial function, we can find the limits of the individual terms of the function, and then add them together. Also, the limit    of a polynomial function as x approaches a is equivalent to simply evaluating the function for a .

Given a function containing a polynomial, find its limit.

  1. Use the properties of limits to break up the polynomial into individual terms.
  2. Find the limits of the individual terms.
  3. Add the limits together.
  4. Alternatively, evaluate the function for a .

Evaluating the limit of a function algebraically

Evaluate lim x 3 ( 5 x 2 ) .

lim x 3 ( 5 x 2 ) = 5 lim x 3 ( x 2 ) Constant times a function property                  = 5 ( 3 2 ) Function raised to an exponent property                  = 45
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Evaluate lim x 4 ( x 3 5 ) .

59

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Evaluating the limit of a polynomial algebraically

Evaluate lim x 5 ( 2 x 3 3 x + 1 ) .

lim x 5 ( 2 x 3 3 x + 1 ) = lim x 5 ( 2 x 3 ) lim x 5 ( 3 x ) + lim x 5 ( 1 ) Sum of functions                                 = 2 lim x 5 ( x 3 ) 3 lim x 5 ( x ) + lim x 5 ( 1 ) Constant times a function                                 = 2 ( 5 3 ) 3 ( 5 ) + 1 Function raised to an exponent                                 = 236 Evaluate
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Evaluate the following limit: lim x 1 ( x 4 4 x 3 + 5 ) .

10

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Finding the limit of a power or a root

When a limit includes a power or a root, we need another property to help us evaluate it. The square of the limit    of a function equals the limit of the square of the function; the same goes for higher powers. Likewise, the square root of the limit of a function equals the limit of the square root of the function; the same holds true for higher roots.

Evaluating a limit of a power

Evaluate lim x 2 ( 3 x + 1 ) 5 .

We will take the limit of the function as x approaches 2 and raise the result to the 5 th power.

lim x 2 ( 3 x + 1 ) 5 = ( lim x 2 ( 3 x + 1 ) ) 5                        = ( 3 ( 2 ) + 1 ) 5                        = 7 5                        = 16,807
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Evaluate the following limit: lim x 4 ( 10 x + 36 ) 3 .

64

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If we can’t directly apply the properties of a limit, for example in lim x 2 ( x 2 + 6 x + 8 x 2 ) , can we still determine the limit of the function as x approaches a ?

Yes. Some functions may be algebraically rearranged so that one can evaluate the limit of a simplified equivalent form of the function.

Finding the limit of a quotient

Finding the limit of a function expressed as a quotient can be more complicated. We often need to rewrite the function algebraically before applying the properties of a limit. If the denominator evaluates to 0 when we apply the properties of a limit directly, we must rewrite the quotient in a different form. One approach is to write the quotient in factored form and simplify.

Questions & Answers

how to understand calculus?
Jenica Reply
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
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
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra 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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