Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
Consider the two functions f1(x) =
x2-4
x-2 and f2(x) = x2 - 2x + 2. Which of
the following is the value of (f1(x) + f2(x)) as x → 2 ?
Options:
(A) 0
(B) 6
(C) 8
(D) ∞
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2026 Q.11)**
**1. Governing Concept**:
- **Topic**: Engineering Mathematics & Domain / Continuity of Rational Functions
- Comparing $f_1(x) = \frac{x^2 - 4}{x - 2}$ and $f_2(x) = x^2 - 2x + 2$:
- For $f_1(x)$: The algebraic expression simplifies to $\frac{(x-2)(x+2)}{x-2} = x + 2$ for all $x \neq 2$. At $x = 2$, $f_1(x)$ has a removable discontinuity and is **undefined** (domain $= \mathbb{R} \setminus \{2\}$).
- For $f_2(x) = x^2 - 2x + 2$: A polynomial defined and continuous everywhere on $\mathbb{R}$. At $x = 2$, $f_2(2) = 2^2 - 2(2) + 2 = 2$.
- While $\lim_{x \to 2} f_1(x) = 4$, $f_1(2)$ does not exist, so $f_1(x) \neq f_2(x)$ at $x = 2$.
**2. Option Analysis**:
- **Option (A) is INCORRECT**: The functions are not identical because their domains differ ($x = 2$ is excluded from the domain of $f_1$).
- **Option (B) is CORRECT**: Accurately recognizes the difference in domain and removable singularity behavior.
- **Option (C) is INCORRECT**: Misidentifies the limit value.
- **Option (D) is INCORRECT**: Fails to account for denominator zero.
**3. Final Answer**:
- **Correct Choice**: **B**
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