Answer
$$\lim\limits_{x\to\infty}[\ln(2+x)-\ln(1+x)]=0$$
Work Step by Step
$$A=\lim\limits_{x\to\infty}[\ln(2+x)-\ln(1+x)]$$$$A=\lim\limits_{x\to\infty}\Bigg[\ln\Big(\frac{2+x}{1+x}\Big)\Bigg]$$$$A=\ln\Bigg[\lim\limits_{x\to\infty}\Big(\frac{2+x}{1+x}\Big)\Bigg]$$
Divide both numerator and denominator by $x$, we have
- The numerator:
$\frac{2+x}{x}=\frac{2}{x}+1$
- The denominator:
$\frac{1+x}{x}=\frac{1}{x}+1$
Therefore, $$A=\ln\Bigg[\lim\limits_{x\to\infty}\Big(\frac{\frac{2}{x}+1}{\frac{1}{x}+1}\Big)\Bigg]$$
$$A=\ln\Bigg[\frac{2\times0+1}{0+1}\Bigg]$$
$$A=\ln1=0$$
