2'3 Calculating Limits Using The Limit Laws

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2.3 - Calculating Limits Using The Limit Laws
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Basic Limit Laws
(a, a)
y x
(a, c)
?
y c
?
?
?
a
a
where n is a positive integer.
where n is a positive integer.
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Limit Laws Generalized
Suppose that c is a constant and the following
limits exist
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Limit Laws Generalized
where n is a positive integer.
where n is a positive integer.
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Examples
Evaluate the following limits. Justify each step
using the laws of limits.
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Direct Substitution Property
If f is a polynomial or a rational function and a
is in the domain of f, then
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Examples
You may encounter limit problems that seem to be
impossible to compute or they appear to not
exist. Here are some tricks to help you evaluate
these limits.

• If f is a rational function or complex
• Eliminate common factors.
• Perform long division.
• Simplify the function (if a complex fraction)
• Find a common denominator.
• If radicals exist, rationalize the numerator or
  denominator.
• If absolute values exist, use one-sided limits
  and the following property.

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Direct Substitution Property
Evaluate the following limits, if they exist.
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Theorem
If f(x) ? g(x) when x is near a (except possibly
at a) and the limits of f and g both exist as x
approaches a, then
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The Squeeze (Sandwich) Theorem
If f(x) ? g(x) ? h(x) when x is near a (except
possibly at a) and
then
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Example
Strategy To begin, bind a part of the function
between two real numbers.
Prove that is
true.