EX3. Evaluate the limit: \( \lim _{x \rightarrow \infty} \frac{\ln \left(x^{2}+1\right)}{\ln \left(x^{3}+2 x\right)} \)

To evaluate the limit \[ L = \lim_{x \rightarrow \infty} \frac{\ln(x^2 + 1)}{\ln(x^3 + 2x)}, \] we start by examining the asymptotic behavior of the logarithms as \( x \) approaches infinity. 1. **For the numerator**: As \( x \to \infty \), \[ x^2 + 1 \sim x^2, \] so \[ \ln(x^2 + 1) \sim \ln(x^2) = 2 \ln(x). \] 2. **For the denominator**: As \( x \to \infty \), \[ x^3 + 2x \sim x^3, \] thus \[ \ln(x^3 + 2x) \sim \ln(x^3) = 3 \ln(x). \] Now substituting these approximations into the limit: \[ L = \lim_{x \rightarrow \infty} \frac{2 \ln(x)}{3 \ln(x)} = \lim_{x \rightarrow \infty} \frac{2}{3} = \frac{2}{3}. \] Therefore, the final result is \[ \boxed{\frac{2}{3}}. \]