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Question

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses fraction numerator x plus 5 over denominator x plus 2 end fraction close parentheses to the power of x plus 3 end exponent equals$

$e^{2}$
$e^{3}$
$e^{4}$
e

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: $e cubed$

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses fraction numerator x plus 5 over denominator x plus 2 end fraction close parentheses to the power of x plus 3 end exponent$
$w h e r e space f left parenthesis x right parenthesis equals open parentheses fraction numerator x plus 5 over denominator x plus 2 end fraction close parentheses space a n d space g left parenthesis x right parenthesis equals left parenthesis x plus 3 right parenthesis n o w space w e space k n o w space t h a t space limit as x rightwards arrow infinity of open parentheses f left parenthesis x right parenthesis close parentheses to the power of g left parenthesis x right parenthesis end exponent equals e to the power of limit as x rightwards arrow infinity of g left parenthesis x right parenthesis left parenthesis f left parenthesis x right parenthesis minus 1 right parenthesis end exponent rightwards double arrow e to the power of limit as x rightwards arrow infinity of left parenthesis x plus 3 right parenthesis open parentheses fraction numerator x plus 5 over denominator x plus 2 end fraction minus 1 close parentheses end exponent rightwards double arrow e to the power of limit as x rightwards arrow infinity of left parenthesis x plus 3 right parenthesis open parentheses fraction numerator x plus 5 minus x minus 2 over denominator x plus 2 end fraction close parentheses end exponent rightwards double arrow e to the power of limit as x rightwards arrow infinity of left parenthesis x plus 3 right parenthesis open parentheses fraction numerator 3 over denominator x plus 2 end fraction close parentheses end exponent rightwards double arrow e to the power of limit as x rightwards arrow infinity of 3 open parentheses fraction numerator 1 plus begin display style 3 over x end style over denominator 1 plus begin display style 2 over x end style end fraction close parentheses end exponent rightwards double arrow e cubed$

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