Download - Lecture 5 limit laws
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Calculating Limits Using the Limit Laws2.3
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Calculating Limits Using the Limit Laws
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Example 1Use the Limit Laws and the graphs of f and g in Figure 1 to evaluate the following limits, if they exist.
Figure 1
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Example 1(a) – SolutionFrom the graphs of f and g we see that
and
Therefore we have
= 1 + 5(–1)
= –4
(by Law 1)
(by Law 3)
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Example 1(b) – SolutionWe see that limx1 f (x) = 2. But limx1 g(x) does not exist because the left and right limits are different:
So we can’t use Law 4 for the desired limit. But we can use Law 4 for the one-sided limits:
The left and right limits aren’t equal, so limx1 [f (x)g(x)] does not exist.
cont’d
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Example 1(c) – SolutionThe graphs show that
and
Because the limit of the denominator
is 0, we can’t use Law 5.
The given limit does not exist because the denominator approaches 0 while the numerator approaches a nonzeronumber.
cont’d
Figure 1
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Power Law
If we use the Product Law repeatedly with g(x) = f (x), we obtain the following law.
Power Law
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Powers & Roots
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Two Basic Limits
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Example
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Calculating Limits – Direct Substitution
Functions with the Direct Substitution Property are called continuous at a.
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Calculating Limits – Finding
Find a function that agrees everywhere but at the “problem point”
When direct substitution yields 0/0.
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Finding Limits - Factoring
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Finding Limits – Factoring - Example
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Finding Limits - Rationalization
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Finding Limits-Rationalization-Ex
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Finding Limits - Fractions
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Finding Limits – Fractions - Example