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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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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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