Question

If $f(x) = \begin{cases} x+1, & x \le 1 \\ 3 - ax^2, & x>1 \end{cases}$ is continuous at $x = 1$, then $a$ is

• A. 1
• B. 2
• C. $-3$
• D. $-2$

Hint:

Continuity condition: LHL $=$ RHL $= f(a)$. Always check all three for piecewise functions.

Answer 1

The Correct Option is A

Step 1: Conceptual Understanding:
For continuity at $x = 1$, the left-hand limit, right-hand limit, and function value must all be equal.
Step 2: Explanation in Detail:
$f(1) = 1 + 1 = 2$.
RHL $= \lim_{x\to1^+}(3-ax^2) = 3-a$.
Continuity: $3 - a = 2 \Rightarrow a = 1$.
Step 3: Therefore, Stating the Final Answer
$a = 1$.