Question

Find the value of b for which function \(\rm f(x) = \left\{\begin{matrix} bx-3 &if &x \geq 3 \\ 5x-9 & if & x< 3 \end{matrix}\right.\) is continuous at x = 3 ?

1. -3
2. 3
3. 6
4. 4

Answer 1

Correct Answer - Option 2 : 3

Concept:

For a function say f, \(\mathop {\lim }\limits_{x \to a} f(x)\) exists

\( ⇒ \mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = \;\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = l = \;\mathop {\lim }\limits_{x \to a} f(x)\), where l is a finite value.

Any function say f is said to be continuous at point say a if and only if \(\mathop {\lim }\limits_{x \to a} f(x) = l = f\left( a \right)\), where l is a finite value.

Calculation:

Given: \(\rm f(x) = \left\{\begin{matrix} bx-3 &if &x \geq 3 \\ 5x-9 & if & x< 3 \end{matrix}\right.\) is continuous at x =3.

AS we know that, if a function f is continuous at point say a then \(\mathop {\lim }\limits_{x \to a} f(x) = l = f\left( a \right)\)

⇒ LHL = lim_x→3^- (5x - 9) = (5 ⋅ 3) - 9 = 6.

⇒ RHL =  limx→3+ (bx - 3) = 3b - 3.

As function is continuous at x = 3 so, LHL = RHL.

⇒ 3b - 3 = 6 ⇒ 3b = 9

⇒ b = 3

Hence, option 2 is correct.