1.2, 1.3 Finding Limits Graphically, Numerically, and Analytically
Informal introduction to limits:
lim f ( x) = L
x !c
“the limit of a function f(x) as x approaches c is L”
As you approach c from the left or from the right, the value of the function becomes arbitrarily close to L.
Limits can be determined by:
• Examining the graph (graphically)
• Examining a table of values (numerically)
• Substituting the value of the function (analytically, we will explore this further in section 1.3)
Example 1:
Analytical Solution:
x 2 ! 3x + 2
lim
x"2
x!2
Numerical Solution:
x
1.9
1.99
1.999
2
2.001
2.01
2.1
Graphical Solution:
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y
.9
.99
.999
Undefined
1.001
1.01
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Example 2:
Analytical Solution:
x
lim
x"0
x +1 !1
Numerical Solution:
x
-0.1
-0.01
-0.001
0
0.001
0.01
0.1
y
1.95
1.99
1.999
Undefined
2.0005
2.005
2.049
Graphical Solution:
Example 3:
Analytical Solution:
$1, x % 2
f ( x) = #
"0, x = 2
Numerical Solution:
x
1.9
1.99
1.99
2
2.001
2.01
2.1
lim f ( x)
x!2
Graphical Solution:
Page 2
y
1
1
1
0
1
1
1
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Limits that do not exist:
In some cases you will not be able to determine the limit of a function. The following are three examples of
limits that do not exist.
Case 1. The function f(x) may approach a different value from the right and from the left as x approaches c.
lim
x!0
x
x
Case 2. The function f(x) increases or decreases without bound as x approaches c.
lim
x!0
1
x2
Case 3. The function f(x) oscillates between two values as x approaches c.
" 1$
lim sin
x!0 #
x%
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More Practice with Analytical Solutions of Limits:
Limit:
1.
General Properties of Limits:
lim b = b
x!c
lim 5 =
x!3
(Hint: Use Direct Substitution whenever possible. You can direct substitute with any polynomial function or
rational function with a non-zero denominator.)
2.
lim 3x 2 =
x!2
lim [b " f (x)] = b lim [ f (x)]
x!c
x!c
3.
2#
&
lim $(5 x + 1) + x ! =
x'0 %
"
lim " f (x)± g(x) $ = lim " f (x) $ + lim " g(x) $
% x!c #
% x!c #
%
x!c #
4.
lim [(cos x)(sin x)] =
x!"
lim # f (x)"g(x) % = lim # f (x) % • lim #"g(x) %
& x!c $
& x!c $
&
x!c $
5.
lim
x 2 " 2x " 5
x!5
x +1
lim
x!c
f (x)
f (x) lim
= x!c
g(x) lim g(x)
x!c
(Hint: Factor and simplify before applying the limit.)
2 x 2 ! 5 x ! 12
x "4
x!4
5.
lim
6.
lim
x "1
x3 !1
x !1
(Hint: Rationalize the numerator or denominator as needed, then take the limit.)
7.
lim
x"0
x +1 !1
x
(Quiz on Summer Review Packet covers limits to this point only.)
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Special Trig Limits:
1 " cos x
=0
x!0
x
lim
1.
sin x
=1
x!0
x
lim
sin 3 x
x !0
x
lim
2. lim
x "0
3 ! 3 cos x
x
cos x tan x
x !0
x
3. lim
sin x
x !0 4 x
4. lim
" sin5x $
5. lim
x!0 # sin2x %
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