Question

Let the function \( f(x) \) be defined as:
\[ f(x) = \begin{cases} A + x, & \text{if } x < 2 \\ 1 + x^2, & \text{if } x \geq 2 \end{cases} \] If the function \( f(x) \) is continuous at \( x = 2 \), the value of \( A \) should be _______.

• A. 2
• B. 2.5
• C. 3
• D. 3.5

Hint:

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

Answer 1

The Correct Option is C

For continuity at \( x = 2 \), the left-hand limit (as \( x \to 2^- \)) must equal the right-hand limit (as \( x \to 2^+ \)).

1. Left-hand limit at \( x = 2 \):
   For \( x < 2 \), \( f(x) = A + x \). Thus, \[ \lim_{x \to 2^-} f(x) = A + 2 \]
2. Right-hand limit at \( x = 2 \):
   For \( x \geq 2 \), \( f(x) = 1 + x^2 \). Thus, \[ \lim_{x \to 2^+} f(x) = 1 + 2^2 = 5 \]

For continuity, these two limits must be equal: \[ A + 2 = 5 \quad \Rightarrow \quad A = 3 \] Therefore, the value of \( A \) should be 3.