p57 section 1.3: evaluating limits analytically if a
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p57 Section 1.3: Evaluating Limits Analytically
If a function is continuous at c you can evaluate the limit by direct substitution. ** You can use the direct substitution method for finding a limit if f is a polynomial or rational function with nonzero denominators That means:
2. 3.
Theorem 1.2: Properties of Limits (these are not worded exactly like your book!)
Theorem 1.1: Some Basic LimitsLet b and c be real numbers and let n be a positive integer.1.
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Theorem 1.3: Limits of Polynomial and Rational FunctionsIf p is a polynomial function and c is a real number, then
If r is a rational function given by and c is a real number
such that q(c) ≠ 0, then
Theorem 1.4: The Limit of a Function Involving a RadicalLet n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even (why?)
Theorem 1.5: The Limit of a Composite FunctionIf f & g are functions such that and then
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Theorem 1.6: Limits of Trigonometric FunctionsLet c be a real number in the domain of the given trigonometric function.1. 2. 3.
4. 5. 6.
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A Strategy for Finding Limits
Theorem 1.7: Functions that Agree at all but One PointLet c be a real number and let f(x) = g(x) for all x ≠ c in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists and
Strategies for Finding Limits1. Learn to recognize which limits can be evaluated through direct
substitution (These limits are listed in Theorems 1.1 through 1.6)2. If the limit of f(x) as x approaches c cannot be evaluated by direct
substitution, try to find a function g that agrees with f for all x other than x = c. (Choose g such that the limit of g(x) can be evaluated by direct substitution).
3. Apply Theorem 1.7 to conclude analytically that
4. Use a graph or table to reinforce your conclusion.
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Dividing Out and Rationalizing Techniques
These techniques give another way to evaluate the limit of a function that cannot be done by direct substitution because it would get 0/0. This is called an indeterminate form because you cannot (from this form alone) determine the limit.
Steps for Dividing Out:1. Try direct substitution first, if you get an indeterminate form,2. Factor everything completely and cancel any common terms3. Use direct substitution again
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Steps for Rationalizing Technique:1. Try direct substitution, if it fails2. Can you factor the numerator? If not,3. Multiply top and bottom by the conjugate of the numerator & simplify4. Use direct substitution again
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Theorem 1.8: The Squeeze TheoremIf h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself, and if
then exists and is equal to L.
** See figure 1.21 in book
Theorem 1.9: Two Special Trigonometric Limits1. 2.
** Proof of theorem 1.9 is on p63
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