103lab6prep.pdf - MA103 Lab Notes Indeterminate Forms and...

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MA103 Lab Notes Indeterminate Forms and L’Hospital’ s Rule (Text: 4.4) The basic indeterminate forms [IFs] are 0 0 and , arising from lim x a f ( x ) g ( x ) . L’Hospital’s Rule states that if a limit expression takes one of these two basic forms, then lim x a f ( x ) g ( x ) = lim x a f 0 ( x ) g 0 ( x ) when the right hand side exists. In some cases, repeated applications of L’Hospital’s Rule may be required and the rule can also be applied to one-sided limits and limits at infinity. Other indeterminate forms are listed below. Each may be converted to one of the basic forms by the method(s) indicated. Then, if necessary, L’Hospital’s Rule may be used. 0 · ∞ Write the product as a quotient: f ( x ) · g ( x ) = f ( x ) 1 /g ( x ) = g ( x ) 1 /f ( x ) . ∞ - ∞ Factor, rationalize, or find a common denominator as appropriate. 0 , 1 , 0 0 Express with base e : f ( x ) g ( x ) = e g ( x ) · ln f ( x ) = exp [ g ( x ) · ln f ( x )]. A common error is to use derivatives when evaluating limits having an indeterminate form which is not one of the basic forms. L’Hospital’s Rule can only be applied to I.F.’s 0 0 and ; but not all limits of these forms can be evaluated by use of the rule. In some cases, techniques previously discussed may have to be employed . Example: Evaluate lim x 1 x ( x - 1) - 1 . lim x 1 x ( x - 1) - 1 ( IF 1 ) = lim x 1 e ln x ( x - 1) - 1 = exp lim x 1 ln x x - 1 IF 0 0 H = exp lim x 1 1 /x 1 = exp(1) = e Maximum and Minimum Function Values (Text: 4.1) Formally: Let f be a function defined on interval I . Then f ( c ), c I , is the absolute maximum value of f if f ( c ) f ( x ) for all x I , and f ( d ), d I , is the absolute minimum value of f if f ( d ) f ( x ) for all x I . The value f ( c ) is said to be a relative [or local ] maximum value of f if there is an open interval I 1 I with c I 1 and f ( c ) f ( x ) for all x I 1 . We define a relative [local] minimum value of f similarly. Graphically: Here is an example function defined on R : Because lim x →∞ f ( x ) = , the function does not have an absolute maximum value. The function does have an absolute minimum value of ’ - 10 occurring at x 3 . 8 since this is the smallest value the function attains on its domain. The function also has a relative maximum value of 2 . 5 at x 0 . 7
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