4.6 Notes - LHopital - Example: x x e x x 2 lim Dont forget...

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Section 4.6 – Indeterminate Forms and L’Hopital’s Rule To evaluate limits of the form 0 0 , 0 , 1 , 0 , , 0 0 we use some form of L’Hopital’s Rule which states: Let f and g be functions that are differentiable on all open intervals (a,b) containing c, except possible at c itself. Assume that 0 ) ( x g for all x in (a,b), except possible at c itself. If the limit of ) ( ) ( x g x f as x approaches c produces the indeterminate form 0 0 , then ) ( ) ( lim ) ( ) ( lim x g x f x g x f c x c x provided the limit on the right exists or is infinite. This result also applies if the limit produces any of the indeterminate forms ) ( ) ( , ) ( , ) ( , . Example: 2 0 sin lim x x x x Example: x x x x cos 1 ln 1 lim 1 L’Hopital’s Rule also applies when you are approaching infinity.
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Unformatted text preview: Example: x x e x x 2 lim Dont forget that you know other ways to evaluate limits! Example: 1 lim 2 x x x Example: 1 3 2 lim 2 1 x x x x Summary of Indeterminate Forms: , Use LHopital directly (possibly more than once) Rewrite as either or then apply LHopital , , 1 Consider the limit of the natural log of the function. Use the properties of logs to rewrite in the form . Rewrite as either or . Apply LHopital Exponentiate your answer Try to rewrite so that you can use one of the previous forms. Example: ) sin )(ln (sin lim x x x Example:...
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4.6 Notes - LHopital - Example: x x e x x 2 lim Dont forget...

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