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Question

$text If end text f left parenthesis 9 right parenthesis equals 9 comma f to the power of apostrophe left parenthesis 9 right parenthesis equals 4 comma text then end text fraction numerator L t over denominator x not stretchy rightwards arrow t end fraction fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction text equals end text$

1
4
0
3

hint Hint:

To find the value of the limit first we will use L hospital theorem, then we will put t=9 to get the answer.

The correct answer is: 4

$f left parenthesis 9 right parenthesis equals 9 space comma space f apostrophe left parenthesis 9 right parenthesis equals 4 limit as x rightwards arrow t of fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction equals limit as x rightwards arrow 9 of fraction numerator square root of f left parenthesis x right parenthesis end root minus 3 over denominator square root of x minus 3 end fraction u sin g space L space h o s p i t a l space t h e o r e m comma equals limit as x rightwards arrow 9 of fraction numerator begin display style fraction numerator f apostrophe left parenthesis x right parenthesis over denominator 2 square root of f left parenthesis x right parenthesis end root end fraction end style minus 0 over denominator begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style minus 0 end fraction equals fraction numerator begin display style fraction numerator f apostrophe left parenthesis 9 right parenthesis over denominator 2 square root of f left parenthesis 9 right parenthesis end root end fraction end style over denominator begin display style fraction numerator 1 over denominator 2 square root of 9 end fraction end style end fraction equals fraction numerator begin display style fraction numerator 4 over denominator 2 square root of 9 end fraction end style over denominator begin display style fraction numerator 1 over denominator 2 square root of 9 end fraction end style end fraction equals 4$

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