Question

What is the value of $\log_{X \to \infty} \left[ -(x + 1) \left( e^{x+1} - 1 \right) \right]$?

• A. Does not exist
• B. 0
• C. -1
• D. 1

Hint:

When analyzing logarithms of functions with large exponential growth, consider the behavior of the terms inside the logarithm as \( x \to \infty \).

Answer 1

The Correct Option is C

We are given the expression: \[ \log_{X \to \infty} \left[ -(x + 1) \left( e^{x+1} - 1 \right) \right] \] Let's analyze the behavior of the given expression as \( x \to \infty \).

Step 1: Analyze the function inside the logarithm.
As \( x \to \infty \), the term \( e^{x+1} \) grows exponentially, so we can approximate the expression \( e^{x+1} - 1 \) by \( e^{x+1} \). This simplifies the expression to: \[ -(x + 1) e^{x+1} \]

Step 2: Take the logarithm.
Now, we apply the logarithm to the simplified expression: \[ \log_{X \to \infty} \left[ -(x + 1) e^{x+1} \right] \] This becomes: \[ \log_{X \to \infty} \left[ -(x + 1) \right] + \log_{X \to \infty} \left[ e^{x+1} \right] \]

Step 3: Simplify further.
As \( x \to \infty \), \( \log_{X \to \infty} \left[ e^{x+1} \right] = x+1 \). So we have: \[ \log_{X \to \infty} \left[ -(x + 1) \right] + (x + 1) \] Now, since \( -(x + 1) \) tends to negative infinity, the logarithm of this term does not exist. Therefore, the value of the given expression tends towards \( -1 \).

Step 4: Conclusion.
The correct answer is \( -1 \), corresponding to option (c).