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1.3limits analytically.notebook 1 October 08, 2012 1.3Evaluating Limits Analytically
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1.3Evaluating Limits Analytically
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Properties of Limits
• In Section 1.2, we learned that the limit of f(x) as x approaches a does not depend on the value of f at x=a.
• However, the limit could be f(a). • In such cases, the limit can be evaluated by direct substitution.
• These types of functions are continuous at a.
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Theorem 1.1Some Basic Limits
Let b and c be real numbers and let n be a positive integer.
1. 2. 3.
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If lim f(x) exists and lim g(x) exists, then the following are true. x a x a
1. [f(x) ± g(x)] =
2. [cf(x)] = c where c is a constant
3. [f(x)g(x)] =
4. f(x) as long as bottom ≠ 0 g(x)
5.
limx a
limx a
f(x) limx a
g(x)±
limx a
limx a
limx a
f(x)
limx a
limx a
f(x) limx a
g(x)
=limx a
f(x)limx a
g(x)
x
Theorem 1.2Properties of LimitsLimit Laws
(Sum or difference)
(Constant multiple)
(Product)
(Quotient)
limx a [f(x)]n= (Power)
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Use the following graph for the next examples.
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Examples:
1. lim [f(x) + 5g(x)] =
2. lim [f(x)g(x)] =
3. lim
4. lim [f(x) g(x)] =
x 2
x 1
x 2 f(x)g(x) =
5. g(1) = 6. f(1) = 7. f(2) =
x 2
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Theorem 1.3Limits of Polynomial and Rational Functions
Assume p and q are polynomials and a is a constant.
a) Polynomial functions:
b) Rational functions: , provided q(a) 0
Theorem 1.4The Limit of a Function Involving a RadicalLet n be a positive integer. The following limit is valid for all a if n is odd and is valid for a > 0 if n is even.
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Theorem 1.5The Limit of a Composite FunctionIf f and g are functions such that and , then
Theorem 1.6Limits of Trigonometric FunctionsLet a be a real number in the domain of the given trigonometric function.
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Theorem 1.7Functions that Agree at All but One PointLet a be a real number and let f(x)=g(x) for all x a in an open interval containing a. If the limit of g(x) as x approaches a exists, then the limit of f(x) also exists and
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Strategies for Finding Limits:1. Try direct substitution. Learn to recognize which limits can be
used by this method.
2. If the limit of f(x) as x approaches a cannot be evaluated by direct substitution, try to find a function g that agrees with f for all x other than x=a.
3. Apply Theorem 1.7 to conclude analytically that
4. Use a graph or table to check also.
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We looked at how to find a limit by "dividing out". What do we do if that doesn't work?
When you try to evaluate a limit by direct substitution and encounter the form 0/0, this is called an indeterminate form because you cannot (from the form alone) determine the limit.
You must rewrite the fraction so that the new denominator does not have 0 as its limit.
We will need to rationalize the numerator.
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More Examples:
1.
2.
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Examples:
3.
4.
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Examples:
5.
6.
7.
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Theorem 1.8-
The Squeeze TheoremIf f(x) ≤ g(x) ≤ h(x) when x is near a
ANDlim f(x) lim h(x) x a x a
THENlim g(x) = L.x a
= = L
f fgg
g
• The limit of a function is squeezed between two other functions, each of which has the same limit at a given xvalue.
• Also called the Sandwich Theorem or the Pinching Theorem
Theorem 1.9Two Special Trig Functions
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More Examples:
1.
2.
