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Question

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript space open parentheses fraction numerator x plus 6 over denominator x plus 1 end fraction close parentheses to the power of x plus 4 end exponent equals$

$e^{6}$
$e^{4}$
e
$e^{5}$

hint 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: $e to the power of 5$

$limit as x rightwards arrow infinity of open parentheses fraction numerator x plus 6 over denominator x plus 1 end fraction close parentheses to the power of open parentheses x plus 4 close parentheses end exponent w h e r e space f left parenthesis x right parenthesis equals open parentheses fraction numerator x plus 6 over denominator x plus 1 end fraction close parentheses space a n d space g left parenthesis x right parenthesis equals left parenthesis X plus 4 right parenthesis 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 open parentheses x plus 4 close parentheses open parentheses fraction numerator x plus 6 over denominator x plus 1 end fraction minus 1 close parentheses end exponent equals e to the power of limit as x rightwards arrow infinity of open parentheses x plus 4 close parentheses open parentheses fraction numerator x plus 6 minus x minus 1 over denominator x plus 1 end fraction close parentheses end exponent equals e to the power of limit as x rightwards arrow infinity of open parentheses x plus 4 close parentheses fraction numerator 5 over denominator x plus 1 end fraction space space space space space space space space end exponent open square brackets space d i v i d i n g 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 space b y space x. close square brackets equals e to the power of limit as x rightwards arrow infinity of fraction numerator 1 plus begin display style 4 over x end style over denominator 1 plus begin display style 1 over x end style end fraction 5 end exponent equals e to the power of 5 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 1 over x to the power of n equals 0 close square brackets$

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