Question Statement:
f(x) = {A + x , if x < 2
1 + x2 , if x ≥ 2
If the function f(x) is continuous at x = 2, the value of A is _____.
Options:
-
(A) 2
-
(B) 2.5
-
(C) 3
-
(D) 3.5
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.15)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given the piecewise function:
$f(x) = \begin{cases} A + x, & x < 2 \\ 1 + x^2, & x \ge 2 \end{cases}$
and given that $f(x)$ is continuous at $x = 2$. We need to solve for parameter $A$.
**2. Step-by-Step Mathematical Derivation**:
1. **Mathematical Condition for Continuity**:
For $f(x)$ to be continuous at $x = 2$, the left-hand limit, right-hand limit, and function value must all be equal:
$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2)$
2. **Left-Hand Limit (LHL)**:
$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (A + x) = A + 2$
3. **Right-Hand Limit & Function Value (RHL)**:
$f(2) = \lim_{x \to 2^+} (1 + x^2) = 1 + (2)^2 = 1 + 4 = 5$
4. **Equating Limits**:
$A + 2 = 5 \implies A = 5 - 2 = 3$
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 2**: ❌ **INCORRECT**. Gives LHL $= 4 \ne 5$.
- **(B) 2.5**: ❌ **INCORRECT**. Gives LHL $= 4.5 \ne 5$.
- **(C) 3**: ✅ **CORRECT**. Substituting $A = 3$ yields $\lim_{x \to 2^-} f(x) = 3 + 2 = 5 = f(2)$, satisfying continuity.
- **(D) 3.5**: ❌ **INCORRECT**. Gives LHL $= 5.5 \ne 5$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **C**
- **Exam Takeaway**: Continuity at a junction point requires $\text{LHL} = \text{RHL} = f(c)$. Set the two expressions equal at the boundary point to solve for the unknown parameter directly.
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