GATE AE 2025 • Q15 MCQ • 1 Mark Engineering Mathematics

Engineering Mathematics

Calculus

Subtopic: Limits, Continuity & Differentiability

Question Statement:

f(x) = {A + x , if x < 2 1 + x^2 , 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

Official Answer Key:

C

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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