Question

• A. \( \frac{9}{5} \)
• B. \( \frac{5}{9} \)
• C. \( \frac{5}{3} \)
• D. \( \frac{3}{5} \)

Hint:

For a piecewise function to be continuous, the limits from both sides at the boundary point must be equal.

Answer 1

The Correct Option is A

Step 1: Continuity condition. 
For the function to be continuous at \( x = 5 \), the left-hand limit and right-hand limit must be equal at \( x = 5 \). 
Step 2: Left-hand limit. 
The left-hand limit is given by \( Kx + 1 \) for \( x \leq 5 \). At \( x = 5 \): \[ \lim_{x \to 5^-} f(x) = K(5) + 1 = 5K + 1 \] Step 3: Right-hand limit. 
The right-hand limit is given by \( 3x - 5 \) for \( x>5 \). At \( x = 5 \): \[ \lim_{x \to 5^+} f(x) = 3(5) - 5 = 15 - 5 = 10 \] Step 4: Equating the limits. 
For continuity at \( x = 5 \), we set the left-hand limit equal to the right-hand limit: \[ 5K + 1 = 10 \] Solving for \( K \): \[ 5K = 9 \quad \Rightarrow \quad K = \frac{9}{5} \] Step 5: Conclusion. 
Thus, the value of \( K \) is \( \frac{9}{5} \), corresponding to option (a).