Question

Given the function:

\[ f(x) = \begin{cases} \frac{(2x^2 - ax +1) - (ax^2 + 3bx + 2)}{x+1}, & \text{if } x \neq -1 \\ k, & \text{if } x = -1 \end{cases} \]

If \( a, b, k \in \mathbb{R} \) and \( f(x) \) is continuous for all \( x \), then the value of \( k \) is:

• A. \( -\frac{1}{3} \)
• B. \( 6 \)
• C. \( a - 2 \)
• D. \( a - 3 \)

Hint:

For continuity at a point \( x = c \), ensure \( \lim_{x \to c} f(x) = f(c) \) by simplifying expressions and canceling terms carefully.

Answer 1

The Correct Option is D

Step 1: Condition for continuity For \( f(x) \) to be continuous at \( x = -1 \), \[ \lim_{x \to -1} f(x) = f(-1). \] Substituting \( x = -1 \) in the numerator, simplifying, and equating to \( k \), we get: \[ k = a - 3. \]