Question

Let \( f(x) = (\ln x)^2, x>0 \). Then

• A. \( \lim_{x \to \infty} f(x) \) does not exist
• B. \( \lim_{x \to \infty} f(x) = 2 \)
• C. \( \lim_{x \to \infty} \left( f(x + 1) - f(x) \right) = 0 \)
• D. \( \lim_{x \to \infty} \left( f(x + 1) - f(x) \right) \) does not exist

Hint:

For large values of \( x \), \( \ln(x + 1) \) can be approximated as \( \ln x + \frac{1}{x} \), helping simplify expressions like \( f(x+1) - f(x) \).

Answer 1

The Correct Option is C

Step 1: Analyze \( f(x) = (\ln x)^2 \) as \( x \to \infty \).
We know that \( \ln x \to \infty \) as \( x \to \infty \), and hence \( f(x) = (\ln x)^2 \to \infty \). Therefore, option (A) is incorrect.

Step 2: Analyzing \( \lim_{x \to \infty} \left( f(x + 1) - f(x) \right) \).
To evaluate the limit of the difference \( f(x + 1) - f(x) \), we use a series expansion for \( \ln(x + 1) \) around \( x \). The difference tends to 0 as \( x \to \infty \). Therefore, option (C) is correct.

Step 3: Conclusion.
The correct answer is \( \boxed{(C)} \).