1. The problem asks to find the limit as $x \to \infty$ of the expression $$\ln(1 + x^2) - \ln(1 + x)$$. 2. Recall the logarithm property: $$\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$$. 3. Apply this property to rewrite the expression: $$\ln\left(\frac{1 + x^2}{1 + x}\right)$$. 4. For large $x$, the dominant terms in numerator and denominator are $x^2$ and $x$ respectively, so: $$\frac{1 + x^2}{1 + x} \approx \frac{x^2}{x} = x$$. 5. Therefore, the expression behaves like $$\ln(x)$$ as $x \to \infty$. 6. Since $$\lim_{x \to \infty} \ln(x) = \infty$$, the limit is: $$\boxed{\infty}$$.