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:
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:
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For continuity at a point \( x = c \), ensure \( \lim_{x \to c} f(x) = f(c) \) by simplifying expressions and canceling terms carefully.
Updated On: Mar 19, 2025
- \( -\frac{1}{3} \)
- \( 6 \)
- \( a - 2 \)
- \( a - 3 \)
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The Correct Option is D
Solution and Explanation
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:
\[
k = a - 3.
\]
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