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Maths-

General

Easy

Question

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

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

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: $e to the power of 4$

$Lt subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses fraction numerator x squared plus 5 x plus 3 over denominator x squared plus x plus 2 end fraction close parentheses to the power of x w h e r e space f left parenthesis x right parenthesis equals open parentheses fraction numerator x squared plus 5 x plus 3 over denominator x squared plus x plus 2 end fraction close parentheses space a n d space g left parenthesis x right parenthesis equals space 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 x squared plus 5 x plus 3 over denominator x squared plus x plus 2 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 x squared plus 5 x plus 3 minus x squared minus x minus 2 over denominator x squared plus x plus 2 end fraction close parentheses end exponent equals e to the power of limit as x rightwards arrow infinity of x. open parentheses fraction numerator 4 x plus 1 over denominator x squared plus x plus 2 end fraction close parentheses end exponent equals e to the power of limit as x rightwards arrow infinity of. open parentheses fraction numerator 4 x squared plus x over denominator x squared plus x plus 2 end fraction close parentheses end exponent equals e to the power of limit as x rightwards arrow infinity of x. open parentheses fraction numerator 4 plus begin display style 1 over x end style over denominator 1 plus begin display style 1 over x end style plus begin display style 2 over x squared end style end fraction close parentheses space space space space space space end exponent open square brackets d i v 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 squared close square brackets equals e to the power of 4$

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