Question

If $\alpha$ is a repeated root of $a{x}^2+bx+c=0$ then $\underset{x\to \alpha }{lim}\frac{\sin (a{x}^2+bx+c)}{{(x-\alpha )}^2}$ is

• A. $0$
• B. $a$
• C. $b$
• D. $c$

Answer 1

Answer: (B)

$a$

$a{(x-\alpha )}^2=a{x}^2+bx+c$

Given $\alpha$ is repeated root and co efficient of $x^2$ should be a

$\Rightarrow {lim}_{x\to \alpha }\frac{\sin (a{x}^2+bx+c)}{{(x-\alpha )}^2}={lim}_{x\to \alpha }\frac{\sin \left(a{(x-\alpha )}^2\right)}{a{(x-\alpha )}^2}a=1.a=a$

As $x\to \alpha$

$a(x-\alpha {)}^2\to 0$