Lesson 8_INDETERMINATE FORMS (L'Hopital's Rule).pptx -...

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INDETERMINATE FORMS
.  2 3 1 3 - x lim 1 x 3 x 1 - x lim 1 x 3 x 4 x lim : follows as numerator the factor we exist, to limit the for and form, ate indetermin an is limit the 0 0 1 1 3 ) 1 ( 4 ) 1 ( 1 x 3 x 4 x lim 1 x 3 x 4 x lim of it lim the Evaluate : call Re 1 x 1 x 2 1 x 2 2 1 x 2 1 x 2 1 x 3 x 4 x lim , thus 2 1 x used. be will Rule s Hopital' L' on Theorems limit said the evaluate To example. second the to applied be longer no can problems previous the in applied principle the Obviously, 0 0 0 ) 0 sin( ) 0 ( 2 ) 0 ( 2 sin 2x 2x sin lim 2x 2x sin lim the evaluating consider us Let 0 x 0 x
. 1 , , 0 5. and - 4. 0 3. : Forms Secondary B. 2. and 0 0 1. : Forms Primary A. : forms ate indetermin of Kinds 0 0
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. EXAMPLE : Evaluate the following limits. 2x 2x sin lim . 1 0 x 0 0 0 0 sin 0 2 0 2 sin 2x 2x sin lim 0 x 1 0 cos 2 0 2 cos 2 1 2 2 x 2 cos lim 2x dx d 2x sin dx d lim 2x 2x sin lim : Rule s ' Hopital ' L g sin u By 0 x 0 x 0 x 1 2x 2x sin lim 0 x
. 3y sin - y 3y - y tan lim . 2 0 y 0 0 0 0 0 0 0 sin3 - 0 0 3 - 0 tan 3y sin - y 3y - y tan lim 0 y 1 2 2 3 1 3 1 0 3cos - 1 3 0 sec 3 3y cos - 1 1 3 y sec lim 3y sin - y dx d 3y - y tan dx d lim 3y sin - y 3y - y tan lim : LHR By 2 2 0 y 0 y 0 y 1 3y sin - y 3y - y tan lim 0 y
. 2 4 x x 4 2x sin ln lim . 3  4 x 4 2 2 x 2 cos 2x sin 1 lim x 4 dx d 2x sin ln dx d lim x 4 2x sin ln lim : LHR By 4 x 2 4 x 2 4 x . ate min er det in still is This 0 0 0 8 2 2cot 4 4 8 4 2 cot 2 x 4 8 2x 2cot lim 4 x 0 0 0 2 sin ln 4 4 4 2 sin ln x 4 2x sin ln lim 2 2 4 x 0 1 ln : Note ln  0 ln
. 32 x 2 csc 4 lim ) 4 ( 8 2 x 2 csc 2 lim x 4 8 dx d 2x cot 2 dx d lim : LHR peat Re 2 4 x 2 4 x 4 x 8 1 1 8 1 4 2 csc 8 1 x 2 csc 8 1 lim 2 2 2 4 x 8 1 x 4 sin2x ln lim 2 4 x
. x 2 x e x lim . 4   e e x lim 2 x 2 x    e 2 1 e 2x lim e dx d x dx d lim e x lim : LHR By x x x 2 x x 2 x 0 2 e 2 1 e 1 2 lim e dx d 2x dx d lim : LHR peat Re x x x x   0 e x lim x 2 x 
. 3x tan ln 3x cos ln lim . 5 6 x - ln 0 ln 2 tan ln 2 cos ln 6 3 tan ln 6 3

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