Limits and Their Properties
Calculus Chapter 1
An Introduction to Limits
Calculus 1.1
Calculus is…
• The mathematics of change
• Velocity
• Acceleration
• The mathematics of tangent lines,
slopes, areas, volumes, arc lengths,
centroids, curvatures, etc., that enable
scientists, engineers, and economists to
model real-life situations.
Calculus Chapter 1
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Calculus is …
•
A limit machine with three stages
1. Precalculus
2. Limit process
3. Calculus formulation
• Derivatives
• Integrals
Calculus Chapter 1
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Tangent Line Problem
• Except for vertical tangent lines, to find
the tangent line you must simply find its
slope
• You already know a point
Calculus Chapter 1
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Secant line
• Used to approximate slope of tangent
line
• A line through the point of tangency (P)
and a second point on the curve (Q)
P   c, f (c) 
Q   c  x, f (c  x) 
Calculus Chapter 1
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Slope of secant line
msec 
f  c  x   f  c 
c  x  c

f  c  x   f  c 
x
• As point Q approaches point P, the
slope of the secant line approaches the
slope of the tangent line.
•  slope of the tangent line is said to be
the limit of the slope of the secant line

Calculus Chapter 1
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Limit
• If f(x) becomes arbitrarily close to a
single number L as x approaches c from
either side, the limit of f(x) as x
approaches c is L
lim f  x   L
x c
Calculus Chapter 1
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Example
x2
lim 2
x2 x  4
• What happens at x = 2?
• To get an idea, look at values close to 2 from the left
and right
x
1.9
1.99
1.999
2.001
2.01
2.1
f(x)
Calculus Chapter 1
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Important to remember
• The existence or nonexistence of f(x) at
x = c has no bearing on the existence of
the limit of f(x) as x approaches c.
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Example
x x  3
lim f  x   
x 3
0 x  3
x
2.9
2.99
2.999 3.001 3.01
3.1
f(x)
Calculus Chapter 1
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Limits that fail to exist
• Behavior that differs from the right and
the left
• Unbounded behavior
• Oscillating behavior
• There are others
Calculus Chapter 1
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Example
lim
x 0
x
x
• If x is positive, f(x) = 1
• If x is negative, f(x) = -1
• No matter how close we get to 0, there will always be
negative 1 on the left and positive 1 on the right
Calculus Chapter 1
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Example
1
lim 2
x 0 x
• As x gets closer to zero from either side, f(x)
gets larger and larger
• “increases without bound”
• Limit does not exist
Calculus Chapter 1
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Example
1
lim sin
x 0
x
x
2/p
2/3p
2/5p
2/7p
2/9p
2/11p
f(x)
Calculus Chapter 1
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Example cont’d
• See page 65
• You can’t always trust the picture your
calculator draws
• It’s wrong, but you can probably still
tell there is not a limit
Calculus Chapter 1
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Note
• When we write
lim f  x   L
x c
• We imply that the limit exists and the limit is L.
• If the limit of a function exists, it is unique.
Calculus Chapter 1
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Properties of Limits
Calculus 1.2
Direct substitution
• Works for some functions
• Called continuous at c
• When
lim f  x   f  c 
x c
• This section – all limits can be evaluated this
way
Calculus Chapter 1
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Basic limits
lim b  b
x c
lim x  c
x c
lim x  c
n
n
x c
Calculus Chapter 1
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You try
lim 6
x 4
lim x
x 12
lim x4
x 2
Calculus Chapter 1
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Properties of limits
• Page 71
• Can be used on all limits, even those
that can’t be evaluated by direct
substitution
Calculus Chapter 1
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Examples


lim  x  1
x 1
2
2
lim
x 3 x  2
Calculus Chapter 1
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Limits with radicals
• Let n be a positive integer. The
following is valid for all c if n is odd, and
is valid for c > 0 if n is even.
lim n x  n c
x c
Calculus Chapter 1
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Limit of a composite function
• If f and g are functions such that
lim g  x   L
x c
lim f  x   f  L 
x L
lim f  g  x    f  L 
x c
Calculus Chapter 1
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Examples
lim f  x   27
x c
• Find
lim 3 f  x 
x c
lim
x c
f  x
18
Calculus Chapter 1
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Limits of trig functions
• All can be evaluated by direct
substitution
• Page 74
Calculus Chapter 1
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Example
2
lim cos x
x 0
Calculus Chapter 1
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Techniques for Evaluating Limits
Calculus 1.3
Indeterminate form
• Direct substitution yields 0/0
• Can’t find limit directly
Calculus Chapter 1
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Functions that agree at all but
one point
• If function is undefined at point c, find
another function that gives the same
values for all other points, and is
defined at point c.
• Cancellation
• Rationalization
Calculus Chapter 1
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Example - cancellation
2 x
lim 2
x2 x  4
Calculus Chapter 1
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You try
2x  x  3
lim
x 1
x 1
2
Calculus Chapter 1
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Example - rationalization
2 x  2
lim
x 0
x
Calculus Chapter 1
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You try
x 1  2
lim
x 3
x3
Calculus Chapter 1
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The Squeeze Theorem
• Page 80
Calculus Chapter 1
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Example
sin x
lim
x 0
x
Calculus Chapter 1
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Example
1  cos x
lim
x 0
x
Calculus Chapter 1
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Two special trig limits
sin x
lim
1
x 0
x
1  cos x
lim
0
x 0
x
Calculus Chapter 1
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Limits with trig functions
• Try to write them using one of the two
special trig forms.
Calculus Chapter 1
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Example
lim
x 0
3 1  cos x 
x
Calculus Chapter 1
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Example
cos x tan x
lim
x 0
x
Calculus Chapter 1
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Example
tan 2 x
lim
x 0
x
Calculus Chapter 1
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Continuity and One-Sided Limits
Calculus 1.4
Continuity
• A function is continuous at x = c if
• There is no interruption in the graph
• The graph is unbroken
• There are no holes, jumps or gaps
Calculus Chapter 1
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Continuity
• A function is continuous at c if
1. f  c  is defined
2. lim f  x  exists
x c
3. lim f  x   f  c 
x c
Calculus Chapter 1
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Continuity over an open interval
• Continuous on open interval if
continuous at each point in the interval
Calculus Chapter 1
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Discontinuities
• When f is not continuous at c, it has a
discontinuity at c
• Removable discontinuity if
• f can be made continuous by defining
or redefining f(c)
• Nonremovable
Calculus Chapter 1
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Discontinuities
• See page 85
• Check for discontinuities where a
function is undefined or in a piecewise
function where the definition of the
function changes
Calculus Chapter 1
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Limit from the right
• x approaches c from values greater
than c
lim f  x   L
x c
Calculus Chapter 1
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Limit from the left
• x approaches c from values less than c
lim f  x   L
x c
Calculus Chapter 1
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One-sided limits
• Useful for taking limits of functions
involving radicals
lim n x  0
x 0
Calculus Chapter 1
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One-sided limits
• Useful for step functions
• See page 86
x  greatest integer n such that n  x
Calculus Chapter 1
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One-sided limits
• Not “official” limits
• If the limit from the left is not equal to
the limit from the right, the limit does not
exist – theorem 1.10
• Unless “from the right” or “from the left”
is specified, it means from both
directions
Calculus Chapter 1
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Continuity over a closed interval
• Continuous over a closed interval [a, b]
if it is continuous over the open interval
(a, b) and
lim  f  a 
x a
lim  f  b 
x b
Calculus Chapter 1
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Example
• Discuss the continuity of
f  x  4  x
Calculus Chapter 1
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56
Properties of Continuity
• Page 89
• Correspond to properties of limits
Calculus Chapter 1
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Functions that are continuous at
every point in their domains
•
•
•
•
Polynomial functions
Rational functions
Radical functions
Trigonometric functions
Calculus Chapter 1
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Continuity of a composite
function
If g is continuous at c
and f is continuous at g  c  ,
then the composite function
f
g  x   f  g  x   is continuous at c.
Calculus Chapter 1
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Testing for continuity
• Describe the intervals on which each
function is continuous
x
f  x   tan
2
f  x  x x  3
Calculus Chapter 1
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Intermediate Value Theorem
• See page 91
• Must be continuous on closed interval
• There may be more than one possibility
Calculus Chapter 1
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Intermediate Value Theorem
• Useful for finding the zeros of a function
• If function is continuous over an
interval and the value of the function
changes in sign over the interval, the
graph must cross y = 0 somewhere in
the interval
Calculus Chapter 1
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Example
• Use the intermediate value theorem to
show that the following function has a
zero on the interval [0, 1]
f  x   x  3x  2
3
Calculus Chapter 1
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Infinite Limits
Calculus 1.5
Infinite limits
• f(x) increases or decreases without
bound as x approaches c
• Approaches a vertical asymptote
• Limit fails to exist
lim f  x   
x c
lim f  x   
x c
Calculus Chapter 1
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Vertical asymptotes
• Occur when the denominator is zero,
but the numerator is not
• Theorem 1.14
• Page 98
Calculus Chapter 1
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Properties of Infinite Limits
• Page 100
Calculus Chapter 1
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Example
• Find the vertical asymptotes (if any) of
the function
f  x 
4
 x  2
3
Calculus Chapter 1
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Example
• Find the vertical asymptotes (if any) of
the function.
x2  4
f  x  3
x  2x2  x  2
Calculus Chapter 1
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Example
• Determine whether the function has a
vertical asymptote or a removable
discontinuity at x = –1
x2  6 x  7
f  x 
x 1
Calculus Chapter 1
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Example
• Find the limit
x3  1
lim 2
x 1 x  x  1
Calculus Chapter 1
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