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Question

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x left parenthesis 1 plus a cos invisible function application x right parenthesis minus b sin invisible function application x over denominator x cubed end fraction equals 1 text then end text straight a equals comma straight b equals$

$\frac{- 5}{2}, \frac{- 3}{2}$
$\frac{5}{2} \frac{3}{2}$
$\frac{2}{3}, \frac{5}{2}$
$\frac{3}{2}, \frac{1}{2}$

hint Hint:

To solve the limit of the function we will have to apply L hospital theorem from which we will get an equation with a and b after that again we have to apply L hospital theorem to get one more equation with a and b, now we can get the value of a and b using the two equations.

The correct answer is: $fraction numerator negative 5 over denominator 2 end fraction comma fraction numerator negative 3 over denominator 2 end fraction$

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x left parenthesis 1 plus a cos x right parenthesis minus b sin x over denominator x cubed end fraction equals 1 L. H. S equals space limit as x rightwards arrow 0 of fraction numerator x left parenthesis 1 plus a cos x right parenthesis minus b sin x over denominator x cubed end fraction u sin g space L space H o s p i t a l space t h e o r e m comma space equals limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator d x over denominator d x end fraction end style left parenthesis 1 plus a cos x right parenthesis plus x. begin display style fraction numerator d open parentheses 1 plus a cos x close parentheses over denominator d x end fraction end style minus begin display style fraction numerator d b sin x over denominator d x end fraction end style over denominator begin display style fraction numerator d x cubed over denominator d x end fraction end style end fraction equals limit as x rightwards arrow 0 of fraction numerator 1 plus a. cos x plus x left parenthesis negative a sin x right parenthesis minus b c s o x over denominator 3 x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator 1 plus a cos x minus a x sin x minus b cos x over denominator 3 x squared end fraction equals limit as x rightwards arrow 0 of fraction numerator 1 plus cos x left parenthesis a minus b right parenthesis minus x left parenthesis a sin x right parenthesis over denominator 3 x squared end fraction n o w comma rightwards double arrow limit as x rightwards arrow 0 of fraction numerator 1 plus cos x left parenthesis a minus b right parenthesis minus x left parenthesis a sin x right parenthesis over denominator 3 x squared end fraction equals 1 rightwards double arrow limit as x rightwards arrow 0 of 1 plus cos x left parenthesis a minus b right parenthesis minus x left parenthesis a sin x right parenthesis equals 3 x squared rightwards double arrow space 1 plus a minus b equals 0 rightwards double arrow a plus 1 equals b......... e q u a t i o n space 1 u sin g space L space h o s p i t a l space t h e o r e m space a g a i n comma space rightwards double arrow limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator d 1 over denominator d x end fraction end style plus left parenthesis a minus b right parenthesis begin display style fraction numerator d cos x over denominator d x end fraction end style minus left parenthesis begin display style fraction numerator d x over denominator d x end fraction end style left parenthesis a sin x space right parenthesis plus x. begin display style fraction numerator d a sin x over denominator d x end fraction end style right parenthesis over denominator 3 begin display style fraction numerator d x squared over denominator d x end fraction end style end fraction rightwards double arrow limit as x rightwards arrow 0 of fraction numerator a minus b left parenthesis negative sin x right parenthesis minus left parenthesis a sin x plus x a cos x right parenthesis over denominator 6 x end fraction rightwards double arrow limit as x rightwards arrow 0 of fraction numerator negative a sin x plus b sin x minus a sin x minus x a cos x over denominator 6 x end fraction rightwards double arrow limit as x rightwards arrow 0 of fraction numerator negative 2 a sin x plus b sin x minus x a cos x over denominator 6 x end fraction rightwards double arrow limit as x rightwards arrow 0 of fraction numerator left parenthesis b minus 2 a right parenthesis sin x over denominator 6 x end fraction minus fraction numerator x a cos x over denominator 6 x end fraction rightwards double arrow limit as x rightwards arrow 0 of fraction numerator left parenthesis b minus 2 a right parenthesis sin x over denominator 6 x end fraction minus limit as x rightwards arrow 0 of fraction numerator a cos x over denominator 6 end fraction n o w comma space rightwards double arrow fraction numerator b minus 2 a over denominator 6 end fraction minus a over 6 equals 1 rightwards double arrow b minus 2 a minus a equals 6 rightwards double arrow b minus 3 a equals 6... e q u a t i o n space 2 b y space e q u a t i o n space 1 space a n d space 2 comma a plus 1 minus 3 a equals 6 1 minus 2 a equals 6 minus 2 a equals 6 minus 1 a equals fraction numerator negative 5 over denominator 2 end fraction b equals fraction numerator negative 5 over denominator 2 end fraction plus 1 equals fraction numerator negative 3 over denominator 2 end fraction$

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