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Question

Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open square brackets fraction numerator a to the power of 1 over x end exponent plus b to the power of 1 over x end exponent plus c to the power of 1 over x end exponent over denominator 3 end fraction close square brackets to the power of xwhere are real and non-zero=

  1. left parenthesis a b c right parenthesis
  2. left parenthesis a b c right parenthesis to the power of 1 third end exponent
  3. left parenthesis a b c right parenthesis to the power of 1 over x end exponent
  4. straight infinity

Hint:

To solve the given function, we will use the formula limit as x rightwards arrow infinity of left parenthesis f left parenthesis x right parenthesis right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals e to the power of stack lim with x rightwards arrow infinity below g left parenthesis x right parenthesis left parenthesis f left parenthesis x right parenthesis minus 1 right parenthesis end exponent after using the formula we will solve the function till that stage when putting the value of x will not make the function undefined, later we will put the value of x and get the required answer.

The correct answer is: left parenthesis a b c right parenthesis to the power of 1 third end exponent


    Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open square brackets fraction numerator a to the power of 1 over x end exponent plus b to the power of 1 over x end exponent plus c to the power of 1 over x end exponent over denominator 3 end fraction close square brackets to the power of x
w h e r e space f left parenthesis x right parenthesis equals open square brackets fraction numerator a to the power of 1 over x end exponent plus b to the power of 1 over x end exponent plus c to the power of 1 over x end exponent over denominator 3 end fraction close square brackets space a n d space g left parenthesis x right parenthesis space equals x
n o w space w e space k n o w space t h a t space limit as x rightwards arrow infinity of open parentheses f left parenthesis x right parenthesis close parentheses to the power of g left parenthesis x right parenthesis end exponent equals e to the power of limit as x rightwards arrow infinity of g left parenthesis x right parenthesis left parenthesis f left parenthesis x right parenthesis minus 1 right parenthesis end exponent
equals e to the power of limit as x rightwards arrow infinity of x. open parentheses fraction numerator a to the power of 1 over x end exponent plus b to the power of 1 over x end exponent plus c to the power of 1 over x end exponent over denominator 3 end fraction minus 1 close parentheses end exponent
equals e to the power of limit as x rightwards arrow infinity of x. open parentheses fraction numerator a to the power of 1 over x end exponent plus b to the power of 1 over x end exponent plus c to the power of 1 over x end exponent minus 3 over denominator 3 end fraction close parentheses end exponent
equals e to the power of limit as x rightwards arrow infinity of x. open parentheses fraction numerator a to the power of 1 over x end exponent minus 1 plus b to the power of 1 over x end exponent minus 1 plus c to the power of 1 over x end exponent minus 1 over denominator 3 end fraction close parentheses end exponent space space space space space space space space space space space space space space space space space space space space space space open square brackets d i v i d i n g space x space f r o m space n u m e r a t o r space a n d space d e n o m i n a t o r close square brackets
equals e to the power of limit as x rightwards arrow infinity of 1 third. open parentheses fraction numerator a to the power of begin display style 1 over x end style end exponent minus 1 over denominator begin display style 1 over x end style end fraction plus fraction numerator b to the power of begin display style 1 over x end style end exponent minus 1 over denominator begin display style 1 over x end style end fraction plus fraction numerator c to the power of begin display style 1 over x end style minus 1 end exponent over denominator begin display style 1 over x end style end fraction close parentheses end exponent
equals e to the power of 1 third open parentheses log space a plus log space b plus log space c close parentheses end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets limit as x rightwards arrow infinity of fraction numerator a to the power of x minus 1 over denominator x end fraction equals log space a close square brackets

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