Question

If \( \lim_{x \to 0} \frac{(1 + a^3)^x + 8e^{1/x}}{1 + (1 - b^3)^x e^{1/x}} = 2 \), then:

• A. \( a = 1, b = 2 \)
• B. \( a = 1, b = -3^{1/3} \)
• C. \( a = 2, b = 3^{1/3} \)
• D. None of these

Hint:

For limits involving exponential and logarithmic terms, simplify the dominant terms first before solving.

Answer 1

The Correct Option is B

Step 1: Simplify the expression

The given limit is: \[ \lim_{x \to 0} \frac{(1 + a^3)^x + 8e^{1/x}}{1 + (1 - b^3)^x e^{1/x}} = 2 \] We will evaluate this limit as \( x \to 0 \). - First, observe that \( e^{1/x} \) grows very quickly as \( x \to 0^+ \). This suggests that \( e^{1/x} \) dominates both the numerator and the denominator for very small values of \( x \). - The term \( (1 + a^3)^x \) can be approximated as follows: \[ (1 + a^3)^x = e^{x \ln(1 + a^3)} \approx 1 \quad \text{as} \quad x \to 0 \] since \( \ln(1 + a^3) \) is a constant and \( x \to 0 \). Similarly, \( (1 - b^3)^x \) can be approximated as: \[ (1 - b^3)^x = e^{x \ln(1 - b^3)} \approx 1 \quad \text{as} \quad x \to 0 \] because \( \ln(1 - b^3) \) is also a constant.

Step 2: Analyze the limit

For very small \( x \), the expression simplifies to: \[ \frac{1 + 8e^{1/x}}{1 + e^{1/x}} = 2 \] As \( x \to 0 \), \( e^{1/x} \) grows very large. So, the expression becomes approximately: \[ \frac{8e^{1/x}}{e^{1/x}} = 8 \] However, we are given that the limit is equal to 2. This means that the numerator and denominator must approach each other in such a way that their ratio is 2. To ensure this, we must adjust the coefficients such that the ratio of the terms balances correctly.

Step 3: Find the values of \( a \) and \( b \)

In order to match the desired limit of 2, we need to determine the values of \( a \) and \( b \) that balance the terms. Given the structure of the problem, a logical assumption would be to set \( a = 1 \) and \( b = 1 \), as these choices will yield the correct balance between the terms. Therefore, we have: \[ a = 1 \quad \text{and} \quad b = 1 \]

Final Answer:

The values of \( a \) and \( b \) are \( \boxed{a = 1} \) and \( \boxed{b = 1} \).