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Maths-

General

Easy

Question

If $L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared$ then the values of a and b are

$a \in R, b \in R$
a=1 and b=2
$a \in R, b = 2$
$a = 1, b \in R$

hint Hint:

In this question we have to find the value of a and b for that we will first solve the limit part, for that we will be using $f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals g left parenthesis x right parenthesis open square brackets f left parenthesis x right parenthesis minus 1 close square brackets$ . Later we will equate both the sides to get the value of a and b.

The correct answer is: $a equals 1 comma b element of R$

$space space space space L t subscript x not stretchy rightwards arrow straight infinity end subscript open parentheses 1 plus a over x plus b over x squared close parentheses to the power of 2 x end exponent equals e squared rightwards double arrow e to the power of limit as x rightwards arrow infinity of 2 x open parentheses 1 plus a over x plus b over x squared minus 1 close parentheses end exponent equals e to the power of 2 space space space space space space space space space end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets u sin g space f left parenthesis x right parenthesis to the power of g left parenthesis x right parenthesis end exponent equals g left parenthesis x right parenthesis open square brackets f left parenthesis x right parenthesis minus 1 close square brackets close square brackets rightwards double arrow limit as x rightwards arrow infinity of 2 x open parentheses a over x plus b over x squared close parentheses equals 2 rightwards double arrow 2 limit as x rightwards arrow infinity of open parentheses a plus b over x close parentheses equals 2 rightwards double arrow 2. left parenthesis a plus 0 right parenthesis equals 2 rightwards double arrow a equals 1 space a n d space b element of R$

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