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If f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator x minus 1 over denominator 2 x to the power of 2 end exponent minus 7 x plus 5 end fraction comma blank f o r x not equal to 1 end cell row cell negative fraction numerator 1 over denominator 3 end fraction comma blank f o r x equals 1 end cell end table close, then f to the power of ´ end exponent left parenthesis 1 right parenthesis is equal to

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  1. negative fraction numerator 1 over denominator 9 end fraction    
  2. negative fraction numerator 2 over denominator 9 end fraction    
  3. negative fraction numerator 1 over denominator 3 end fraction    
  4. fraction numerator 1 over denominator 3 end fraction    

    Answer:The correct answer is: negative fraction numerator 2 over denominator 9 end fractionGiven, f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator x minus 1 over denominator 2 x to the power of 2 end exponent minus 7 x plus 5 end fraction comma blank x not equal to 1 end cell row cell negative fraction numerator 1 over denominator 3 end fraction comma blank x equals 1 end cell end table close
    f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator 1 over denominator 2 x minus 5 end fraction comma blank x not equal to 1 end cell row cell negative fraction numerator 1 over denominator 3 end fraction comma blank x equals 1 end cell end table close
    f to the power of ´ end exponent open parentheses 1 close parentheses equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator f open parentheses 1 plus h close parentheses minus f left parenthesis 1 right parenthesis over denominator h end fraction
    equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator fraction numerator 1 over denominator 2 open parentheses 1 plus h close parentheses minus 5 end fraction minus open parentheses negative fraction numerator 1 over denominator 3 end fraction close parentheses over denominator h end fraction
    equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator fraction numerator 1 over denominator 2 h minus 3 end fraction plus fraction numerator 1 over denominator 3 end fraction over denominator h end fraction equals stack l i m with h rightwards arrow 0 below fraction numerator 3 plus 2 h minus 3 over denominator 3 h left parenthesis 2 h minus 3 right parenthesis end fraction equals negative fraction numerator 2 over denominator 9 end fraction
    L f to the power of ´ end exponent open parentheses 1 close parentheses equals stack l i m with h rightwards arrow 0 below fraction numerator f open parentheses 1 minus h close parentheses minus f left parenthesis 1 right parenthesis over denominator negative h end fraction
    equals stack l i m with h rightwards arrow 0 below fraction numerator fraction numerator 1 over denominator 2 open parentheses 1 minus h close parentheses minus 5 end fraction minus open parentheses negative fraction numerator 1 over denominator 3 end fraction close parentheses over denominator negative h end fraction
    equals stack l i m with h rightwards arrow 0 below minus fraction numerator 2 over denominator 3 left parenthesis 2 h plus 3 right parenthesis end fraction equals negative fraction numerator 2 over denominator 9 end fraction
    therefore blank f to the power of ´ end exponent open parentheses 1 close parentheses equals negative fraction numerator 2 over denominator 9 end fraction

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