LIMITSOF
FUNCTIONS
DEFINITION: LIMITS The most basic use of limits is to describe how a function behaves as the independent variable approaches a given value. For example let us examine the behavior of the function for x-values closer and closer to 2. It is evident from the graph and the table in the next slide that the values of f(x) get closer and closer to 3 as the values of x are selected closer and closer to 2 on either the left or right side of 2. We describe this by saying that the “limit of is 3 as x approaches 2 from either side,” we write
1xx)x(f 2
1xx)x(f 2
31xxlim 2
2x
2
3
f(x)
f(x)
x
y
1xxy 2
x 1.9 1.95 1.99 1.995 1.999 2 2.001 2.005 2.01 2.05 2.1
F(x) 2.71 2.852 2.97 2.985 2.997 3.003 3.015 3.031 3.152 3.31
left side right side
O
This leads us to the following general idea.
EXAMPLEUse numerical evidence to make a conjecture about the value of .
1x1xlim
1x
Although the function is undefined at x=1, this has no bearing on the limit. The table shows sample x-values approaching 1 from the left side and from the right side. In both cases the corresponding values of f(x) appear to get closer and closer to 2, and hence we conjecture that and is consistent with the graph of f.
1x1x)x(f
21x
1xlim1x
x .99 .999 .9999 .99999 1 1.00001 1.0001 1.001 1.01
F(x) 1.9949 1.9995 1.99995 1.999995 2.000005 2.00005 2.0005 2.004915
THEOREMS ON LIMITS
Our strategy for finding limits algebraically has two parts:•First we will obtain the limits of some simpler function•Then we will develop a list of theorems that will enable us to use the limits of simple functions as building blocks for finding limits of more complicated functions.
We start with the following basic theorems, which are illustrated in Fig 1.2.1
axlim b kklim a numbers. real be k and a Let Theorem 1.2.1
axax
33lim 33lim 33lim example, For
a. of values all for ax as kf(x) why explains whichvaries, x as k at fixed remain
f(x) of values the then function, constant a is k xf If
x0x-25x
Example 1.
xlim 2xlim 0xlim
example, For . axf that true be also must it ax then x, xf If
x-2x0x
Example 2.
The following theorem will be our basic tool for finding limits algebraically.
This theorem can be stated informally as follows:
a) The limit of a sum is the sum of the limits.b) The limit of a difference is the difference of the limits.c) The limits of a product is the product of the limits.d)The limits of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.e) The limit of the nth root is the nth root of the limit.
•A constant factor can be moved through a limit symbol.
5x2lim .14x
12x6lim .23x
)2x5(x4lim .33x
EXAMPLE : Evaluate the following limits.
3158
5)4(2
5limxlim2
5limx2lim
4x4x
4x4x
612-18
12)3(6
12limx6lim3x3x
13
131 2)3(534
2limxlim5xlim4lim
2limx5limxlim4lim
2x5limx4lim
3x3x3x3x
3x3x3x3x
3x3x
4x5x2lim .4
5x
3
3x6x3lim .5
3x1x8lim .6
1x
2110
42552
4limxlim5
x lim2
4limx5lim
x2 lim
5x5x
5x
5x5x
5x
337515633
6limxlim3
6limx3lim
6x3lim
33
3
3x3x
3
3x3x
3
3x
23
49
3x1x8lim
1x
OR
When evaluating the limit of a function at a given value, simply replace the variable by the indicated limit then solve for the value of the function:
22
3lim 3 4 1 3 3 4 3 1
27 12 138
xx x
EXAMPLE: Evaluate the following limits.
2x8xlim .1
3
2x
Solution:
00
088
2282
2x8xlim
33
2x
Equivalent function:
(indeterminate)
2x
4x2x2xlim2
2x
124444222
4x2xlim2
2
2x
122x8xlim
3
2x
Note: In evaluating a limit of a quotient which reduces to , simplify the fraction. Just remove the common factor in the numerator and denominator which makes the quotient . To do this use factoring or rationalizing the numerator or denominator, wherever the radical is.
00
00
x22xlim .2
0x
Solution:
Rationalizing the numerator:
(indeterminate)00
0220
x22xlim
0x
22xx22xlim
22x22x
x22xlim
0x0x
42
221
221
22x1lim
22xxxlim
0x0x
42
x22xlim
0x
9x427x8lim .3 2
3
23x
Solution:
By Factoring:
(indeterminate)32
3
3
22
38 278 27 27 27 02lim4 9 9 9 034 9
2
x
xx
3
232
9236
234
3x29x6x4lim
3x23x29x6x43x2lim
2
2
23x
2
23x
223
23
29
627
33999
223
9x427x8lim 2
3
23x
5x3x2xlim .4 2
3
2x
Solution:
33
222
2 2 2 32 3lim5 2 5
8 4 34 5
159
153
x
x xx
315
5x3x2xlim 2
3
2x
DEFINITION: One-Sided Limits
The limit of a function is called two-sided limit if it requires the values of f(x) to get closer and closer to a number as the values of x are taken from either side of x=a. However some functions exhibit different behaviors on the two sides of an x-value a in which case it is necessary to distinguish whether the values of x near a are on the left side or on the right side of a for purposes of investigating limiting behavior.
Consider the function
0x ,10x ,1
xx
)x(f
1
-1
As x approaches 0 from the right, the values of f(x) approach a limit of 1, and similarly , as x approaches 0 from the left, the values of f(x) approach a limit of -1.
1xx
lim and 1xx
lim
,symbols In
oxox
This leads to the general idea of a one-sided limit
EXAMPLE:
xx
)x(f 1. Find if the two sided limits exist given
1
-1
exist. not does xx
lim or
exist not does itlim sided two the thenxx
limxx
lim the cesin
1xx
lim and 1xx
lim
ox
oxox
oxox
SOLUTION
EXAMPLE:2. For the functions in Fig 1.1.13, find the one-sided limit and the two-sided limits at x=a if they exists.
The functions in all three figures have the same one-sided limits as , since the functions are Identical, except at x=a.
ax
1)x(flim and 3)x(flimare itslim These
axax
In all three cases the two-sided limit does not exist as because the one sided limits are not equal. ax
SOLUTION
3. Find if the two-sided limit exists and sketch the graph of
2
6+x if x < -2( ) =
x if x -2g x
4 26
x6lim)x(glim.a2x2x
4
2-
xlim)x(glim.b
2
2
2x2x
4)x(glim or4 to equal is and exist itlim sided two the then
)x(glim)x(glim the cesin
2x
2x2x
SOLUTION
EXAMPLE:
x-2-6 4
y
4
4. Find if the two-sided limit exists and sketch the graph of
2
2
3 + x if x < -2( ) = 0 if x = -2
11 - x if x > -2
f x
SOLUTION
7 23
x3lim)x(flim.a
2
2
2x2x
7 2-11
x11lim)x(flim.b
2
2
2x2x
7)x(flim or7 to equal is and exist itlim sided two the then
)x(flim)x(flimthe cesin
2x
2x2x
EXAMPLE:
graph. the sketch and
,exist f(x) lim if eminerdet ,4x23)x(f If .52x
3
4223
4x23lim )x(flim .a2x2x
3
4223
4x23lim )x(flim .b2x2x
3)x(flim or3 to equal is and exist itlimsided two the then
)x(flim)x(flimthe cesin
2x
2x2x
SOLUTION
EXAMPLE:
f(x)
x
(2,3)
2
DEFINITION: LIMITS AT INFINITY
The behavior of a function as x increases or decreases without bound is sometimes called the end behavior of the function.
)x(f
If the values of the variable x increase without bound, then we write , and if the values of x decrease without bound, then we write .
xx
For example ,
0x1lim and 0
x1lim
xx
x
x
0x1lim
x
0
x1lim
x
In general, we will use the following notation.
Fig.1.3.2 illustrates the end behavior of the function f when L)x(flim or L)x(flim
xx
EXAMPLEFig.1.3.2 illustrates the graph of . As suggested by this graph,
x
x11y
ex11lim
and ex11lim
x
x
x
x
EXAMPLE
6x32xlim .4
x311x2x5lim .3
5x2xx4lim .2
8x65x3lim .1
2
x
23
x
3
2
x
x
336
x
36
x
xx5xlim .6
x5xlim .5