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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

how fast can i understand functions without much difficulty
Joe Reply
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
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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