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Access the following online resource for additional instruction and practice with properties of limits.

Key concepts

  • The properties of limits can be used to perform operations on the limits of functions rather than the functions themselves. See [link] .
  • The limit of a polynomial function can be found by finding the sum of the limits of the individual terms. See [link] and [link] .
  • The limit of a function that has been raised to a power equals the same power of the limit of the function. Another method is direct substitution. See [link] .
  • The limit of the root of a function equals the corresponding root of the limit of the function.
  • One way to find the limit of a function expressed as a quotient is to write the quotient in factored form and simplify. See [link] .
  • Another method of finding the limit of a complex fraction is to find the LCD. See [link] .
  • A limit containing a function containing a root may be evaluated using a conjugate. See [link] .
  • The limits of some functions expressed as quotients can be found by factoring. See [link] .
  • One way to evaluate the limit of a quotient containing absolute values is by using numeric evidence. Setting it up piecewise can also be useful. See [link] .

Section exercises

Verbal

Give an example of a type of function f whose limit, as x approaches a , is f ( a ) .

If f is a polynomial function, the limit of a polynomial function as x approaches a will always be f ( a ) .

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When direct substitution is used to evaluate the limit of a rational function as x approaches a and the result is f ( a ) = 0 0 , does this mean that the limit of f does not exist?

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What does it mean to say the limit of f ( x ) , as x approaches c , is undefined?

It could mean either (1) the values of the function increase or decrease without bound as x approaches c , or (2) the left and right-hand limits are not equal.

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Algebraic

For the following exercises, evaluate the limits algebraically.

lim x 2 ( 5 x x 2 1 )

10 3

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lim x 2 ( x 2 5 x + 6 x + 2 )

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lim x 3 ( x 2 9 x 3 )

6

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lim x 1 ( x 2 2 x 3 x + 1 )

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lim x 3 2 ( 6 x 2 17 x + 12 2 x 3 )

1 2

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lim x 7 2 ( 8 x 2 + 18 x 35 2 x + 7 )

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lim x 3 ( x 2 9 x 5 x + 6 )

6

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lim x 3 ( 7 x 4 21 x 3 12 x 4 + 108 x 2 )

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lim x 3 ( x 2 + 2 x 3 x 3 )

does not exist

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lim h 0 ( ( 3 + h ) 3 27 h )

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lim h 0 ( ( 2 h ) 3 8 h )

12

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lim h 0 ( ( h + 3 ) 2 9 h )

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lim h 0 ( 5 h 5 h )

5 10

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lim x 0 ( 3 x 3 x )

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lim x 9 ( x 2 81 3 x )

108

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lim x 1 ( x x 2 1 x )

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lim x 0 ( x 1 + 2 x 1 )

1

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lim x 1 2 ( x 2 1 4 2 x 1 )

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lim x 4 ( x 3 64 x 2 16 )

6

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lim x 2 ( | x 2 | x 2 )

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lim x 2 + ( | x 2 | x 2 )

1

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lim x 2 ( | x 2 | x 2 )

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lim x 4 ( | x 4 | 4 x )

1

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lim x 4 + ( | x 4 | 4 x )

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lim x 4 ( | x 4 | 4 x )

does not exist

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lim x 2 ( 8 + 6 x x 2 x 2 )

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For the following exercise, use the given information to evaluate the limits: lim x c f ( x ) = 3 , lim x c g ( x ) = 5

lim x c [ 2 f ( x ) + g ( x ) ]

6 + 5

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lim x c [ 3 f ( x ) + g ( x ) ]

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lim x c f ( x ) g ( x )

3 5

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For the following exercises, evaluate the following limits.

lim x 2 cos ( π x )

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lim x 2 sin ( π x )

0

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lim x 2 sin ( π x )

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

3

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

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

does not exist; right-hand limit is not the same as the left-hand limit.

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lim x 4 x + 5 3 x 4

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lim x 2 + ( 2 x 〚x〛 )

2

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lim x 2 x + 7 3 x 2 x 2

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lim x 3 + x 2 x 2 9

Limit does not exist; limit approaches infinity.

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

f ( x ) = 2 x 2 1

4 x + 2 h

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f ( x ) = x 2 + 4 x 100

2 x + h + 4

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f ( x ) = cos ( x )

cos ( x + h ) cos ( x ) h

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f ( x ) = 1 x

1 x ( x + h )

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f ( x ) = x

1 x + h + x

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Graphical

Find an equation that could be represented by [link] .

Graph of increasing function with a removable discontinuity at (2, 3).
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Find an equation that could be represented by [link] .

Graph of increasing function with a removable discontinuity at (-3, -1).

f ( x ) = x 2 + 5 x + 6 x + 3

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For the following exercises, refer to [link] .

Graph of increasing function from zero to positive infinity.

What is the right-hand limit of the function as x approaches 0?

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What is the left-hand limit of the function as x approaches 0?

does not exist

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Real-world applications

The position function s ( t ) = 16 t 2 + 144 t gives the position of a projectile as a function of time. Find the average velocity (average rate of change) on the interval [ 1 , 2 ] .

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The height of a projectile is given by s ( t ) = 64 t 2 + 192 t Find the average rate of change of the height from t = 1 second to t = 1.5 seconds.

52

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The amount of money in an account after t years compounded continuously at 4.25% interest is given by the formula A = A 0 e 0.0425 t , where A 0 is the initial amount invested. Find the average rate of change of the balance of the account from t = 1 year to t = 2 years if the initial amount invested is $1,000.00.

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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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