Single Variable Calculus, Chapter 7, 7.3-1, Section 7.3-1, Problem 50

Determine the limit $\displaystyle \lim_{x \to \infty} [\ln (2 + x) - \ln (1 + x)]$

$
\begin{equation}
\begin{aligned}
\lim_{x \to \infty} [\ln (2 + x) - \ln (1 + x)] &= \lim_{x \to \infty} \ln \left( \frac{2+x}{1+x} \right)\\
\\
\lim_{x \to \infty} [\ln (2 + x) - \ln (1 + x)] &= \lim_{x \to \infty} \ln \left( \frac{\frac{2}{x}+ \frac{x}{x}}{\frac{1}{x}+\frac{x}{x}} \right)\\
\\
\lim_{x \to \infty} [\ln (2 + x) - \ln (1 + x)] &= \lim_{x \to \infty} \ln \left( \frac{\frac{2}{x}+1}{\frac{1}{x}+1} \right)\\
\\
\lim_{x \to \infty} [\ln (2 + x) - \ln (1 + x)] &= \ln \left( \frac{0+1}{0+1}\right)\\
\\
\lim_{x \to \infty} [\ln (2 + x) - \ln (1 + x)] &= \ln \left( \frac{1}{1}\right)\\
\\
\lim_{x \to \infty} [\ln (2 + x) - \ln (1 + x)] &= \ln 1\\
\\
\lim_{x \to \infty} [\ln (2 + x) - \ln (1 + x)] &= 0
\end{aligned}
\end{equation}
$