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Graphical

For the following exercises, estimate the functional values and the limits from the graph of the function f provided in [link] .

A piecewise function with discontinuities at x = -2, x = 1, and x = 4.

lim x 2 f ( x )

–4

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lim x 2 + f ( x )

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lim x 2 f ( x )

–4

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lim x 1 f ( x )

2

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lim x 1 + f ( x )

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lim x 1 f ( x )

does not exist

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lim x 4 f ( x )

4

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lim x 4 + f ( x )

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lim x 4 f ( x )

does not exist

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For the following exercises, draw the graph of a function from the functional values and limits provided.

lim x 0 f ( x ) = 2 , lim x 0 + f ( x ) = 3 , lim x 2 f ( x ) = 2 , f ( 0 ) = 4 , f ( 2 ) = 1 , f ( 3 )  does not exist .

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lim x 2 f ( x ) = 0 , lim x 2 + = 2 , lim x 0 f ( x ) = 3 , f ( 2 ) = 5 , f ( 0 )

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

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lim x 3 f ( x ) = 0 , lim x 3 + f ( x ) = 5 , lim x 5 f ( x ) = 0 , f ( 5 ) = 4 , f ( 3 )  does not exist .

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lim x 4 f ( x ) = 6 , lim x 6 + f ( x ) = 1 , lim x 0 f ( x ) = 5 , f ( 4 ) = 6 , f ( 2 ) = 6

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lim x 3 f ( x ) = 2 , lim x 1 + f ( x ) = 2 , lim x 3 f ( x ) = 4 , f ( 3 ) = 0 , f ( 0 ) = 0

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lim x π f ( x ) = π 2 , lim x π f ( x ) = π 2 , lim x 1 f ( x ) = 0 , f ( π ) = 2 , f ( 0 )  does not exist .

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For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as x approaches 0.

g ( x ) = ( 1 + x ) 2 x

7.38906

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i ( x ) = ( 1 + x ) 4 x

54.59815

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Based on the pattern you observed in the exercises above, make a conjecture as to the limit of f ( x ) = ( 1 + x ) 6 x , g ( x ) = ( 1 + x ) 7 x , and  h ( x ) = ( 1 + x ) n x .

e 6 403.428794 , e 7 1096.633158 , e n

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For the following exercises, use a graphing utility to find 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.

( x ) = { | x | 1 , if  x 1 x 3 , if  x = 1   a = 1

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( x ) = { 1 x + 1 , if  x = 2 ( x + 1 ) 2 , if  x 2   a = 2

lim x 2 f ( x ) = 1

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Numeric

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 near x = a . Round answers to two decimal places.

f ( x ) = x 2 4 x 16 x 2 ; a = 4

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f ( x ) = x 2 x 6 x 2 9 ; a = 3

lim x 3 ( x 2 x 6 x 2 9 ) = 5 6 0.83

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f ( x ) = x 2 6 x 7 x 2   7 x ; a = 7

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f ( x ) = x 2 1 x 2 3 x + 2 ; a = 1

lim x 1 ( x 2 1 x 2 3 x + 2 ) = 2.00

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f ( x ) = 1 x 2 x 2 3 x + 2 ; a = 1

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f ( x ) = 10 10 x 2 x 2 3 x + 2 ; a = 1

lim x 1 ( 10 10 x 2 x 2 3 x + 2 ) = 20.00

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f ( x ) = x 6 x 2 5 x 6 ; a = 3 2

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f ( x ) = x 4 x 2 + 4 x + 1 ; a = 1 2

lim x 1 2 ( x 4 x 2 + 4 x + 1 ) does not exist. Function values decrease without bound as x approaches –0.5 from either left or right.

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f ( x ) = 2 x 4 ;   a = 4

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For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as x approaches the given value.

lim x 0 7 tan x 3 x

lim x 0 7 tan x 3 x = 7 3

Table shows as the function approaches 0, the value is 7 over 3 but the function is undefined at 0.
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lim x 0 2 sin x 4 tan x

lim x 0 2 sin x 4 tan x = 1 2

Table shows as the function approaches 0, the value is 1 over 2, but the function is undefined at 0.
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For the following exercises, use a graphing utility to find 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.

lim x 0 e e   1 x 2

lim x 0 e e   1 x 2 = 1.0

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lim x 1 | x + 1 | x + 1

lim x 1 | x + 1 | x + 1 = ( x + 1 ) ( x + 1 ) = 1 and lim x 1 + | x + 1 | x + 1 = ( x + 1 ) ( x + 1 ) = 1 ; since the right-hand limit does not equal the left-hand limit, lim x 1 | x + 1 | x + 1 does not exist.

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lim x 5 | x 5 | 5 x

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lim x 1 1 ( x + 1 ) 2

lim x 1 1 ( x + 1 ) 2 does not exist. The function increases without bound as x approaches 1 from either side.

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lim x 1 1 ( x 1 ) 3

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lim x 0 5 1 e 2 x

lim x 0 5 1 e 2 x does not exist. Function values approach 5 from the left and approach 0 from the right.

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Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar: f ( x ) = | 1 x x | and g ( x ) = | 1 + x x | as x approaches 0. Use a graphing utility, if possible, to determine the left- and right-hand limits of the functions f ( x ) and g ( x ) as x approaches 0. If the functions have a limit as x approaches 0, state it. If not, discuss why there is no limit.

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Extensions

According to the Theory of Relativity, the mass m of a particle depends on its velocity v . That is

m = m o 1 ( v 2 / c 2 )

where m o is the mass when the particle is at rest and c is the speed of light. Find the limit of the mass, m , as v approaches c .

Through examination of the postulates and an understanding of relativistic physics, as v c , m . Take this one step further to the solution,

lim v c m = lim v c m o 1 ( v 2 / c 2 ) =
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Allow the speed of light, c , to be equal to 1.0. If the mass, m , is 1, what occurs to m as v c ? Using the values listed in [link] , make a conjecture as to what the mass is as v approaches 1.00.

v m
0.5 1.15
0.9 2.29
0.95 3.20
0.99 7.09
0.999 22.36
0.99999 223.61
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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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