Question

Function f is defined as: \[ f(x) = \begin{cases} 2x + 2, & x<2 \\ k, & x = 2 \\ 3x, & x>2. \end{cases} \text{Find the value of } k \text{ for which } f \text{ is continuous at } x = 2. \]

Hint:

For continuity at $x = a$: LHL = RHL = f(a).

Answer 1

Step 1: For continuity at $x = 2$, \[ \lim_{x \to 2^-} f(x) = f(2) = \lim_{x \to 2^+} f(x). \] Step 2: Left-hand limit (LHL): \[ \lim_{x \to 2^-} f(x) = 2(2) + 2 = 4 + 2 = 6. \] Right-hand limit (RHL): \[ \lim_{x \to 2^+} f(x) = 3(2) = 6. \] Step 3: So, \[ f(2) = k = 6. \] Therefore, $k = 6$. %Quciktip