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Finding Limits: Properties of Limits

For the following exercises, find the limits if lim x c f ( x ) = −3 and lim x c g ( x ) = 5.

lim x c ( f ( x ) + g ( x ) )

2

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lim x c f ( x ) g ( x )

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lim x c ( f ( x ) g ( x ) )

−15

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lim x 0 + f ( x ) , f ( x ) = { 3 x 2 + 2 x + 1 5 x + 3    x > 0 x < 0

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lim x 0 f ( x ) , f ( x ) = { 3 x 2 + 2 x + 1 5 x + 3    x > 0 x < 0

3

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lim x 3 + ( 3 x 〚x〛 )

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For the following exercises, evaluate the limits using algebraic techniques.

lim h 0 ( ( h + 6 ) 2 36 h )

12

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lim x 25 ( x 2 625 x 5 )

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lim x 1 ( x 2 9 x x )

10

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lim x 4 7 12 x + 1 x 4

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lim x 3 ( 1 3 + 1 x 3 + x )

1 9

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Continuity

For the following exercises, use numerical evidence to determine whether the limit exists at x = a . If not, describe the behavior of the graph of the function at x = a .

f ( x ) = 2 x 4 ;   a = 4

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f ( x ) = 2 ( x 4 ) 2 ;   a = 4

At x = 4 , the function has a vertical asymptote.

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f ( x ) = x x 2 x 6 ;   a = 3

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f ( x ) = 6 x 2 + 23 x + 20 4 x 2 25 ;   a = 5 2

removable discontinuity at a = 5 2

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f ( x ) = x 3 9 x ;   a = 9

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For the following exercises, determine where the given function f ( x ) is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

f ( x ) = x 2 2 x 15

continuous on ( , )

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f ( x ) = x 2 2 x 15 x 5

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f ( x ) = x 2 2 x x 2 4 x + 4

removable discontinuity at x = 2. f ( 2 ) is not defined, but limits exist.

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f ( x ) = x 3 125 2 x 2 12 x + 10

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f ( x ) = x 2 1 x 2 x

discontinuity at x = 0 and x = 2. Both f ( 0 ) and f ( 2 ) are not defined.

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f ( x ) = x + 2 x 2 3 x 10

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f ( x ) = x + 2 x 3 + 8

removable discontinuity at x = 2.   f ( 2 ) is not defined.

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Derivatives

For the following exercises, find the average rate of change f ( x + h ) f ( x ) h .

f ( x ) = ln ( x )

ln ( x + h ) ln ( x ) h

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For the following exercises, find the derivative of the function.

Find the equation of the tangent line to the graph of f ( x ) at the indicated x value.
f ( x ) = x 3 + 4 x ; x = 2.

y = 8 x + 16

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For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

Given that the volume of a right circular cone is V = 1 3 π r 2 h and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of π

12 π

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Practice test

For the following exercises, use the graph of f in [link] .

Graph of a piecewise function with two segments. The first segment goes from negative infinity to (-1, 0), an open point, and the second segment goes from (-1, 3), an open point, to positive infinity.

lim x −1 + f ( x )

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lim x −1 f ( x )

0

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lim x −2 f ( x )

−1

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At what values of x is f discontinuous? What property of continuity is violated?

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For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as x approaches a . If the function has a limit as x approaches a , state it. If not, discuss why there is no limit

f ( x ) = { 1 x 3 ,  i f x 2 x 3 + 1 , i f x > 2    a = 2

lim x 2 f ( x ) = 5 2 a and lim x 2 + f ( x ) = 9 Thus, the limit of the function as x approaches 2 does not exist.

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f ( x ) = { x 3 + 1 , i f x < 1 3 x 2 1 , i f x = 1 x + 3 + 4 , i f x > 1    a = 1

For the following exercises, evaluate each limit using algebraic techniques.

lim x −5 ( 1 5 + 1 x 10 + 2 x )

1 50

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lim h 0 ( h 2 + 25 5 h 2 )

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lim h 0 ( 1 h 1 h 2 + h )

1

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For the following exercises, determine whether or not the given function f is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

f ( x ) = x 3 4 x 2 9 x + 36 x 3 3 x 2 + 2 x 6

removable discontinuity at x = 3

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For the following exercises, use the definition of a derivative to find the derivative of the given function at x = a .

f ( x ) = 3 x

f ' ( x ) = 3 2 a 3 2

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For the graph in [link] , determine where the function is continuous/discontinuous and differentiable/not differentiable.

Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.

discontinuous at –2,0, not differentiable at –2,0, 2.

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For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

f ( x ) = | x 2 | | x + 2 |

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f ( x ) = 2 1 + e 2 x

not differentiable at x = 0 (no limit)

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For the following exercises, explain the notation in words when the height of a projectile in feet, s , is a function of time t in seconds after launch and is given by the function s ( t ) .

s ( 2 )

the height of the projectile at t = 2 seconds

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s ( 2 ) s ( 1 ) 2 1

the average velocity from t = 1  to  t = 2

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For the following exercises, use technology to evaluate the limit.

lim x 0 sin ( x ) 3 x

1 3

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lim x 0 tan 2 ( x ) 2 x

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lim x 0 sin ( x ) ( 1 cos ( x ) ) 2 x 2

0

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Evaluate the limit by hand.

lim x 1 f ( x ) ,  where   f ( x ) = { 4 x 7 x 1 x 2 4 x = 1

At what value(s) of x is the function below discontinuous?

f ( x ) = { 4 x 7 x 1 x 2 4 x = 1

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For the following exercises, consider the function whose graph appears in [link] .

Graph of a positive parabola.

Find the average rate of change of the function from x = 1  to  x = 3.

2

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Find all values of x at which f ' ( x ) = 0.

x = 1

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Find all values of x at which f ' ( x ) does not exist.

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Find an equation of the tangent line to the graph of f the indicated point: f ( x ) = 3 x 2 2 x 6 ,    x = 2

y = 14 x 18

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For the following exercises, use the function f ( x ) = x ( 1 x ) 2 5 .

Graph the function f ( x ) = x ( 1 x ) 2 5 by entering f ( x ) = x ( ( 1 x ) 2 ) 1 5 and then by entering f ( x ) = x ( ( 1 x ) 1 5 ) 2 .

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Explore the behavior of the graph of f ( x ) around x = 1 by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at x = 1.

The graph is not differentiable at x = 1 (cusp).

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For the following exercises, find the derivative of each of the functions using the definition: lim h 0 f ( x + h ) f ( x ) h

f ( x ) = 4 x 2 7

f ' ( x ) = 8 x

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f ( x ) = 1 x + 2

f ' ( x ) = 1 ( 2 + x ) 2

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f ( x ) = x 3 + 1

f ' ( x ) = 3 x 2

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f ( x ) = x 1

f ' ( x ) = 1 2 x 1

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Questions & Answers

what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
Practice Key Terms 7

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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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