Question

If \( f(x) = \left\{ \begin{array}{ll} 1 + \left| \sin x \right|, & \text{for } -\pi \leq x<0
e^{x/2}, & \text{for } 0 \leq x<\pi
\end{array} \right. \) then the value of \( a \) and \( b \), if \( f \) is continuous at \( x = 0 \), are respectively

• A. \( a = 3, b = e^3 \)
• B. \( a = 2, b = e^3 \)
• C. \( a = 3, b = 2 \)
• D. \( a = 1, b = 2 \)

Hint:

For continuity at a point, the left-hand and right-hand limits must be equal at that point.

Answer 1

The Correct Option is B

Step 1: Applying continuity conditions.
For the function to be continuous at \( x = 0 \), the values of \( a \) and \( b \) must satisfy the condition \( \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) \). Using this condition, we solve for \( a = 2 \) and \( b = e^3 \).

Step 2: Conclusion.
The values of \( a \) and \( b \) are \( a = 2 \) and \( b = e^3 \), corresponding to option (2).