Chapter 3.1 Notes
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Chapter 3: Limits and Continuity
3.1 Limits
The limit of f ( x) as x approaches a is L if we can make the values of
f ( x) arbitrarily close to L as x is close to a but not equal to a.
lim f ( x) = L
x a
When x approaches a from the left, we call this the left-hand limit,
lim- f ( x) .
x a
When x approaches a from the right, we call this the right-hand limit,
lim+ f ( x) .
x a
What can we say about y = f ( x)
at x  1
at x = 1?
lim f ( x) = L if and only if lim- f ( x) = L and lim+ f ( x) = L
x a
x a
x a
Chapter 3.1 Notes
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Let f be a function defined on both sides of a, except possibly at a itself.
Then lim f ( x) = ¥ means that the values of f ( x) can be made arbitrarily
x a
large by taking x sufficiently close to a.
Similarly, lim f ( x) = -¥ when the values of f ( x) can be made
x a
arbitrarily negatively large by taking x sufficiently close to a.
The line x = a is called a vertical asymptote of the function f ( x) if at
least one of the following is true:
lim f ( x) = ¥ or -¥
x a-
lim f ( x) = ¥ or -¥
x a+
Example: Using the graph of f ( x) find the following limits, if they exit
5
f ( x)
c) lim f ( x)
x-2
d) lim f ( x )
-5
5
x
x 0
e) lim f ( x )
x1
a) lim f ( x)
x-5
f) lim f ( x )
x 4
b) lim f ( x)
x-3
Chapter 3.1 Notes
Example:
function
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g
Sketch the graph of the
ìï2 - x
ïï
ïï x
g ( x) = í
ïï4
ïï
ïî4 - x
if x <-1
if -1 £ x < 1
2
if x = 1
if x > 1
2
x
Use the graph to find the following limits, if they exist:
a) lim g ( x)
x-1
b) lim g ( x )
x1
Example: Evaluate the function g ( x) at the given values and then guess
the value of lim+ g ( x) .
x 2
1- x 2
g ( x) = 2
x + 3x -10
g (3) =
g (2.001) =
g (2.1) =
g (2.0001) =
g (2.01) =
g (2.00001) =
Chapter 3.1 Notes
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Example: Evaluate the function g ( x) at the given values and then guess
the value of lim- g ( x) .
x0
g ( x) =
cos x -1
sin x
g (-1) =
g (-0.1) =
g (-0.5) =
g (-0.01) =
g (-0.3) =
g (-0.001) =
Example: Find the infinite limits
a) lim
x0
x -1
x 2 ( x + 2)
6
x 5 5 - x
b) lim
Chapter 3.1 Notes
5
Example: Find the vertical asymptotes of y =
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x
and sketch the
x2 - x - 2
graph.
5
-5
5
Calculating Limits Using the Limit Laws
Suppose that c is a constant, n is a positive integer, and the limits
lim f ( x) and lim g ( x)
x a
x a
exist. Then the following laws can be used to calculate limits.
Note that n is a positive integer
1.
lim[ f ( x)  g ( x) ] = lim f ( x)  lim g ( x)
2.
lim[cf ( x)] = c lim f ( x)
3.
lim[ f ( x) ⋅ g ( x) ] = lim f ( x) ⋅ lim g ( x)
4.
f ( x)
f ( x) lim
x a
lim
if lim g ( x) ¹ 0
=
x a g ( x )
x a
lim g ( x)
x a
x a
x a
x a
x a
x a
x a
x a
x a
Chapter 3.1 Notes
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The following can also be used when finding limits:
lim [ f ( x ) ] = éêlim f ( x )ùú
x a
ë x a
û
n
n
lim c = c
x a
lim x n = a n
x a
lim n x = n a (if n is even, assume a > 0 )
x a
lim n f ( x) = n lim f ( x) (if n is even, assume lim f ( x) > 0 )
x a
x a
x a
Example: Given that lim f ( x ) = 3 , find the following using the limit laws.
x 2
2
a) lim éê( f ( x )) - 2 x + 4ùú
x 2 ë
û
f ( x) + 6
x 2 2 f ( x ) + x + 11
b) lim
Chapter 3.1 Notes
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Example: Find the following limits, if they exist
a) lim ( x 2 + x + 1)
5
x-2
x 2 - x -12
b) lim
x-3
x +3
c) lim
x64
(
3
x +3 x
x2 - x - 3
d) lim
x-1
x +1
)
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Chapter 3.1 Notes
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(2 + h) - 8
3
e) lim
h 0
f) lim +
x-1.5
h
(
3 + 2x + x
)
x +1
x 4 ( x - 4) 2
g) lim
t 3 - t 2 - t + 10
h) lim 2
t -2
t + 3t + 2
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