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Question

L t subscript y not stretchy rightwards arrow 0 end subscript fraction numerator sin invisible function application left parenthesis a plus b x right parenthesis minus sin invisible function application left parenthesis a minus b x right parenthesis over denominator x end fraction

  1. 2 b cos space a
  2. 2 b sin space a
  3. 2 cos space a
  4. 2 sin space a

Hint:

In the question first we will use L hospital theorem. Then we can put the value of x to get the answer.

The correct answer is: 2 b cos space a


    L t subscript y not stretchy rightwards arrow 0 end subscript fraction numerator sin invisible function application left parenthesis a plus b x right parenthesis minus sin invisible function application left parenthesis a minus b x right parenthesis over denominator x end fraction
    using L hospital theorem,
    limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator d sin left parenthesis a plus b x right parenthesis over denominator d x end fraction end style minus begin display style fraction numerator d sin left parenthesis a minus b x right parenthesis over denominator d x end fraction end style over denominator begin display style fraction numerator d x over denominator d x end fraction end style end fraction
equals limit as x rightwards arrow 0 of fraction numerator b. cos space left parenthesis a plus b x right parenthesis plus b. cos space left parenthesis a minus b x right parenthesis over denominator 1 end fraction
equals b. cos left parenthesis a plus b.0 right parenthesis plus b. cos left parenthesis a minus b.0 right parenthesis
equals b. cos space a plus b. cos a
equals 2 b. cos a

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