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Limits and Their Properties 1

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A Preview of Calculus

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1.1

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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’d Precalculus concepts

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Precalculus concepts cont’d

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Precalculus concepts cont’d

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Finding Limits Graphically

and Numerically

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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?

(Class example)

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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 1.5.

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.”

Figure 1.5

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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

(See figure 1.10 on page 51)

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Solution:

Consider the graph of the function . From

Figure 1.8 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

Figure 1.8

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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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Evaluating Limits Analytically

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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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• Let’s do some algebra…

An expression such as 0/0 is called

an indeterminate form because

you cannot (from the form alone)

determine the limit. (When you try

to plug in x = 1, you get the 0/0.)

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Because exists, you can apply Theorem 1.7 to

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 Figure 1.17

Figure 1.17

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.

• 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 the limit of the numerator is also 0, the numerator

and denominator have a common factor of (x + 3).

So, for all x ≠ –3, you can divide out this factor to obtain

It follows that:

Example 7 – Solution cont’d

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This result is shown graphically in Figure 1.18.

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 – Solution

Figure 1.18

cont’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

Solution:

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In this case, you can rewrite the fraction by rationalizing the

numerator.

cont’d Example 8 – Solution

(Continued on next page)

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Now, using Theorem 1.7, you can evaluate the limit

as shown.

cont’d Example 8 – Solution

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A table or a graph can reinforce your conclusion that the

limit is . (See Figure 1.20.)

Figure 1.20

Example 8 – Solution cont’d