Evaluating Limits Analytically

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Evaluating Limits Analytically

• Section 1.3

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After this lesson, you will be able to

• evaluate a limit using the properties of limits
• develop and use a strategy for finding limits
• evaluate a limit using dividing out and
  rationalizing techniques

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Limits? Analytically
In the previous lesson, you learned how to find
limits numerically and graphically. In this
lesson you will be shown how to find them
analyticallyusing algebra or calculus.
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Theorem 1.1 Some Basic Limits
Let b and c be real numbers and let n be a
positive integer.
Examples
Think of it graphically
y scale was adjusted to fit
As x approaches 5, f(x) approaches 125
As x approaches 3, f(x) approaches 4
As x approaches 2, f(x) approaches 2
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Direct Substitution

• Some limits can be evaluated by direct
  substitution for x.
• Direct substitution works on continuous
  functions.
• Continuous functions do NOT have any holes,
  breaks or gaps.

Note Direct substitution is valid for all
polynomial functions and rational functions whose
denominators are not zero.
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Theorem 1.2 Properties of Limits
Let b and c be real numbers, let n be a positive
integer, and let f and g be functions with the
following limits
Scalar multiple
Sum or difference
Product
Quotient
Power
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Limit of a Polynomial Function
Example
Since a polynomial function is a continuous
function, then we know the limit from the right
and left of any number will be the same. Thus,
we may use direct substitution.
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Limit of a Rational Function
Make sure the denominator doesnt 0 !
Example
If the denominator had been 0, we would not have
been able to use direct substitution.
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Theorem 1.4 The Limit of a Function Involving a
Radical
Let n be a positive integer. The following limit
is valid for all c if n is odd, and is valid for
c gt 0 if n is even.
So we can use direct substitution again, as long
as c is in the domain of the radical function.
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Theorem 1.5 The Limit of a Composite Function
If f and g are functions such that lim g(x) L
and lim f(x) f(L), then
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Limit of a Composite Function-part a
Example Given
and ,
find
a) First find
Direct substitution works here
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Limit of a Composite Function -part b
Direct substitution works here, too.

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Limits of Trig Functions
If c is in the domain of the given trig function,
then
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Limits of Trig Functions
Examples
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Limits of Trig Functions
Examples
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Finding Limits

• Try Direct Substitution
• If the limit of f(x) as x approaches c cannot be
  evaluated by direct substitution, try to divide
  out common factors or to rationalize the
  numerator so that direct substitution works.
• Use a graph or table to reinforce your result.

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Example 1- Factoring
Example
Graph on your calculator and use the table to
check your result
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Example 2- Factoring
Example
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Example Limit DNE
Example
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Example 1- Rationalizing Technique
Example
First, we will try direct substitution
Plan Rationalize the numerator to come up with a
related function that is defined at x 0.
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Example 1- Rationalizing Technique
Example
Go ahead and graph to verify.
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Example 2- Rationalizing Technique
Example
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Two Very Important Trig Limits
(A star will indicate the need to memorize!!!)
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Example 1- Using Trig Limits
Example
Now, the 5x is like the heart. You will need the
bottom to also be 5x in order to use the trig
limit. So, multiply the top and bottom by 5.
You wont have changed the fraction. Watch how
to do it.
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Example 2
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Example 3
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Example 4
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Example 5
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Homework
Section 1.3 page 67 1, 5-39 odd, 49-61 odd,
67-77 odd