Question

The value of \( \lim_{x \to \infty} \dfrac{x \ln(x)}{1 + x^2} \) is:

• A. 0
• B. 1.0
• C. 0.5
• D. \( \infty \)

Hint:

When evaluating limits, consider the growth rates of terms in the numerator and denominator. The term with the higher power of \( x \) will dominate as \( x \to \infty \).

Answer 1

The Correct Option is A

We are asked to evaluate the limit: \[ \lim_{x \to \infty} \frac{x \ln(x)}{1 + x^2}. \] As \( x \to \infty \), the denominator grows much faster than the numerator because \( x^2 \) dominates \( \ln(x) \). Therefore, the limit of this expression as \( x \to \infty \) is 0. 

Final Answer: \[ \boxed{0}. \]