Question

Assertion :$\underset{x\to \beta }{lim}{(1+a{x}^2+bx+c)}^{\frac{1}{x-\beta }}$ is equal to $e^{a(\alpha -\beta )}$ Reason: If $\alpha$, $\beta$ are roots of $a{x}^2+bx+c=0$ then $a{x}^2+bx+c=a(x-\alpha )(x-\beta )$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$\because$ $\alpha$, $\beta$ are roots of $a{x}^2+bx+c=0$
$\therefore$ $a{x}^2+bx+c=a(x-\alpha )(x-\beta )$
$\therefore$ $\underset{x\to \beta }{lim}{(1+a{x}^2+bx+c)}^{\frac{1}{x-\beta }}$
$=\underset{x\to \beta }{lim}{[1+a(x-\alpha )(x-\beta )]}^{\frac{1}{x-\beta }}$
$=e^{{lim}_{x\to \beta }[1+a(x-\alpha )(x-\beta )-1]\times \frac{1}{x-\beta }}$
$=e^{a(\beta -\alpha )}$

Therefore both assertion and reason are correct, but reason is not correct explanation for assertion.