GATE BT 2024 • Q63
NAT • 2 Marks
Section 2: General Biotechnology
Biomolecules, enzyme kinetics
Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
The value of the limit lim
x→∞
x
2 ln(1 +
2024
x ) is ________.
Official Answer Key:
1012 to 1012
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2024 Q.63)**
**1. Governing Concept & Definition**:
- **Topic**: Calculus & Limits of Indeterminate Forms
- Evaluating limits using standard logarithmic series expansion:
$\ln(1 + u) = u - rac{u^2}{2} + \dots \quad ext{as } u
ightarrow 0$
**2. Detailed Step-by-Step Calculation**:
- **Expression**:
$\lim_{x o \infty} rac{x}{2} \ln\left(1 + rac{2024}{x}
ight)$
- **Substitution**:
Let $u = rac{2024}{x}$. As $x o \infty$, $u o 0$.
$\ln\left(1 + rac{2024}{x}
ight) pprox rac{2024}{x}$
- **Evaluating Limit**:
$\lim_{x o \infty} \left[ rac{x}{2} \cdot rac{2024}{x}
ight] = rac{2024}{2} = 1012$
**3. Final Answer**:
- **Calculated Value**: **1012**
- **Official GATE Accepted Range**: **1012 to 1012**
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