Answer
$0$
Work Step by Step
Find limit $\lim\limits_{x \to \infty}[ln(2+x)-ln(1+x)]$
Consider $[ln(2+x)-ln(1+x)]$
Use logarithmic property, $lnx-lny=ln\frac{x}{y}$
Thus, $[ln(2+x)-ln(1+x)]=ln[\frac{2+x}{1+x}]$
Now we will find the limit.
$\lim\limits_{x \to \infty}[ln(2+x)-ln(1+x)]=\lim\limits_{x \to \infty}[ln[\frac{2+x}{1+x}]$
or
$\lim\limits_{x \to \infty}[ln[\frac{2+x}{1+x}]=\lim\limits_{x \to \infty}[ln\frac{\frac{2}{x}+1}{\frac{1}{x}+1}]$
Since ${x \to \infty}$ then ${\frac{2}{x} \to \ 0}$ and
${\frac{1}{x} \to \ 0}$
Hence, $\lim\limits_{x \to \infty}[ln(2+x)-ln(1+x)]=\lim\limits_{x \to \infty}ln[\frac{0+1}{0+1}]=ln1 =0$
