Question

If $\underset{x\to 1}{lim}\frac{x^2-ax+b}{x-1}=5,$ then $a+b$ is equal to :

• A. $1$
• B. $5$
• C. $-4$
• D. $-7$

Answer 1

The correct option is D $-7$

Given
$\underset{x\to 1}{lim}\frac{x^2-ax+b}{x-1}=5$
In order to limit exist, numerator must be equal to $0$ at $x=1$
$\Rightarrow (1{)}^2-a(1)+b=0$
$1-a+b=0$
$a-b=1...\left(1\right)$

Now, applying L'hospital Rule
$\underset{x\to 1}{lim}\frac{2x-a}{1}=5$
$\Rightarrow 2-a=5$
$\Rightarrow a=-3...\left(2\right)$
putting the value of $a$ in $\left(1\right)$
$\therefore b=-4$
$\Rightarrow a+b=-7$