Consider the function, where [ x ] denotes the greatest integer less than or equal to x . If S denotes the set of all ordered pairs ( a , b ) such that f ( x ) is continuous at x = 3 , then the number of elements in S is: [20

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Solution

Q.
Consider the function,

$f (x)$ $= {\begin{matrix} \frac{a (7 x - 12 - x^{2})}{b | x^{2} - 7 x + 12 |}, & x < 3 \\ 2^{\frac{\sin (x - 3)}{x - [x]},} & x > 3 \\ b, & x = 3 \end{matrix}$

where $[x]$ denotes the greatest integer less than or equal to $x .$ If $S$ denotes the set of all ordered pairs $(a, b)$ such that $f (x)$ is continuous at $x = 3,$ then the number of elements in $S$ is: [2024]

1 2

2 Infinitely many

3 1

4 4

Ans.
(3)

$As f (x) is continuous at x = 3 .$

$So, LHL=RHL = f (a)$

$\Rightarrow \lim_{x \to 3^{-}} \frac{a (7 x - 12 - x^{2})}{b | x^{2} - 7 x + 12 |} = \lim_{x \to 3^{+}} 2^{\frac{\sin (x - 3)}{x - [x]}} = b$

$\Rightarrow \frac{- a}{b} = 2^{1} = b$ $[∵ \lim_{n \to 0} \frac{\sin n}{n} = 1]$

$\Rightarrow b = 2, a = - 4 ∴ Number of elements in S = 1$

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