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Limits and Their Properties1
Copyright © Cengage Learning. All rights reserved.
A Preview of Calculus
Copyright © Cengage Learning. All rights reserved.
1.1
3
What Is Calculus?
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Calculus is the mathematics of change. For instance,
calculus is the mathematics of velocities, accelerations,
tangent lines, slopes, areas, volumes, arc lengths,
centroids, curvatures, and a variety of other concepts that
have enabled scientists, engineers, and economists to
model real-life situations.
Although precalculus mathematics also deals with
velocities, accelerations, tangent lines, slopes, and so on,
there is a fundamental difference between precalculus
mathematics and calculus.
Precalculus mathematics is more static, whereas calculus
is more dynamic.
Calculus
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Precalculus concepts
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cont’dPrecalculus concepts
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Precalculus conceptscont’d
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Precalculus conceptscont’d
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Finding Limits Graphically
and Numerically
Copyright © Cengage Learning. All rights reserved.
1.2
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Estimate a limit using a numerical or
graphical approach. (GNAW on Calculus)
Learn different ways that a limit can fail to
exist.
Objectives
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An Introduction to Limits
What is a limit?
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Suppose you are asked to sketch the graph of the function
f given by
For all values other than x = 1, you can use standard
curve-sketching techniques.
However, at x = 1, it is not clear what to expect.
An Introduction to Limits
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An Introduction to Limits
• To get an idea of the behavior of the graph
of f near x = 1, you can use two sets of x-
values–one set that approaches 1 from the
left and one set that approaches 1 from the
right, as shown in the table.
x 0.75 1 1.25
f(x)
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To get an idea of the behavior of the graph of f near x = 1,
you can use two sets of x-values–one set that approaches
1 from the left and one set that approaches 1 from the right,
as shown in the table.
An Introduction to Limits
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The graph of f is a parabola that
has a gap at the point (1, 3), as
shown in the figure at the right.
Although x can not equal 1, you
can move arbitrarily close to 1,
and as a result f(x) moves
arbitrarily close to 3.
Using limit notation, you can write
An Introduction to Limits
This is read as “the limit of f(x) as x approaches 1 is 3.”
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An Introduction to Limits
This discussion leads to an informal definition of 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.
This limit is written as
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Limits That Fail to Exist
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Solution:
Consider the graph of the function . From the
graph shown and the definition of absolute value
Example 3 – Behavior That Differs from the Right and from the Left
Show that the limit does not exist.
you can see that
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Limits That Fail to Exist
What about f(x) increasing or decreasing
without bound as x approaches c?
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Limits That Fail to Exist
What about f(x) increasing or decreasing
without bound as x approaches c?
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Does the limit exist?
DNE
2
2
2
0
DNE
-3
0
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• Finding limits for piece-wise functions:
Find -4
-5
DNE
-13
Plug -1 into top two equations &
see if they give the same value.
Plug 0 into middle equation.
Plug 3 into bottom two equations
& see if they give the same value.
Plug -4 into top equation.
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Evaluating Limits Analytically
Copyright © Cengage Learning. All rights reserved.
1.3
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Develop and use a strategy for finding limits.
Evaluate a limit using dividing out (factor and
cancel) and rationalizing techniques.
Objectives
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Remember this function?
Could we have figured
out the limit as x
approaches 1 without
graphing or
doing a table?
An Introduction to Limits
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Because exists, you can conclude that f and g
have the same limit at x = 1.
Example 6 – Solution cont’d
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Example 6 – Solution
So, for all x-values other than x = 1, the functions f and g
agree, as shown in the figures below.
cont’d
f and g agree at all but one point
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Limits
• So
• But what about:
or
= 1
= 3
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Limits• The first thing we do when finding limits
is to try plugging in the x to see what y
value we get. (See pages 59-61 for
properties & example.)
• If you can’t plug in the x, then try doing
some algebra and then see if you can
plug in the x, (factor & cancel, or
rationalize).
• If that doesn’t work, use a graph or table
to determine the limit.
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Example 1 – Evaluating Basic Limits
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Find the limit:
Example 3 – The Limit of a Rational Function
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Example 4(a) – The Limit of a Composite Function
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Example 4(b) – The Limit of a Composite Function
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Example 5 – Limits of Trigonometric Functions
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Example 7 – Dividing Out (factor & cancel)
Find the limit:
What happens to the numerator and the denominator when
you plug in the -3?
It is an indeterminate form because
you cannot (from the form alone)
determine the limit. (When you plug
in x = -3, you get the fraction 0/0
which is undefined.)
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Because you get a 0 when you plug in -3 in both the
numerator and denominator, then they have a common
factor. For this example it is (x + 3).
So, for all x ≠ –3, you can divide out this factor to obtain
It follows that:
Example 7 – Solutioncont’d
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This result is shown graphically in the figure below.
Note that the graph of the function f coincides with the
graph of the function g(x) = x – 2, except that the graph of f
has a gap at the point (–3, –5).
Example 7 – Solutioncont’d
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Rationalizing Technique
(another way to simplify)
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Find the limit:
(By direct substitution, you obtain the indeterminate form 0/0.
Example 8 – Rationalizing Technique
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In this case, you can rewrite the fraction by rationalizing the
numerator.
cont’dExample 8 – Solution
(Continued on next page)
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Now you can evaluate the limit as shown.
cont’dExample 8 – Solution
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A table or a graph can reinforce your conclusion that the
limit is .
Example 8 – Solutioncont’d
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Review:• The first thing we do when finding limits
is to try plugging in the x to see what y
value we get.
• If you can’t plug in the x, then try doing
some algebra and then see if you can
plug in the x, (factor & cancel, or
rationalize).
• If that doesn’t work, use a graph or table
to determine the limit.
