f(x)= ll x^2-9 x-3 +a , &x>3 5 , &x=3 2 x^2+3 x+b, &x<3 . is continuous at x=3, then

f(x)= ll x^2-9 x-3 +a , &x>3 5 , &x=3 2 x^2+3 x+b, &x<3 . is cont…

MCQ

$f(x)=\left\{\begin{array}{ll}\frac{x^2-9}{x-3}+a , &x>3 \\ 5 , &x=3 \\ 2 x^2+3 x+b, &x<3\end{array}\right.$
is continuous at $x=3$, then

A
$a=1, b=-22$
B
$a=1, b=22$
C
$a=-1, b=22$
✓
$a=-1, b=-22$

✓

Answer

Correct option: D.

$a=-1, b=-22$

(D)
Since $f (x)$ is continuous at $x=3$.
$\therefore \quad \lim _{x \rightarrow 3^{-}} f (x)=\lim _{x \rightarrow 3^{+}} f (x)= f (3)$
$\lim _{x \rightarrow 3^{-}} f (x)= f (3)$
$\Rightarrow \lim _{x \rightarrow 3^{-}}\left(2 x^2+3 x+ b \right)=5$
$\Rightarrow 2(3)^2+3(3)+b=5$
$\Rightarrow b =-22$
Also, $\lim _{x \rightarrow 3^{+}} f (x)= f (3)$
$\Rightarrow \lim _{x \rightarrow 3^{+}}\left(\frac{x^2-9}{x-3}+ a \right)=5$
$\Rightarrow(3+3+a)=5$
$\Rightarrow a =-1$

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