find the limit. lim ln(4 + x) - ln(1 + x) x→∞

Explanation:

Step1: Use logarithm property

By the property $\ln a-\ln b = \ln\frac{a}{b}$, we have $\lim_{x\rightarrow\infty}[\ln(4 + x)-\ln(1 + x)]=\lim_{x\rightarrow\infty}\ln\frac{4 + x}{1 + x}$.

Step2: Simplify the fraction

Divide both the numerator and denominator of $\frac{4 + x}{1 + x}$ by $x$: $\frac{4 + x}{1 + x}=\frac{\frac{4}{x}+1}{\frac{1}{x}+1}$.

Step3: Find the limit of the fraction

As $x\rightarrow\infty$, $\lim_{x\rightarrow\infty}\frac{\frac{4}{x}+1}{\frac{1}{x}+1}=\frac{\lim_{x\rightarrow\infty}\frac{4}{x}+\lim_{x\rightarrow\infty}1}{\lim_{x\rightarrow\infty}\frac{1}{x}+\lim_{x\rightarrow\infty}1}$. Since $\lim_{x\rightarrow\infty}\frac{4}{x}=0$ and $\lim_{x\rightarrow\infty}\frac{1}{x}=0$, we get $\frac{0 + 1}{0+1}=1$.

Step4: Find the limit of the logarithm

Since $\lim_{x\rightarrow\infty}\frac{4 + x}{1 + x}=1$, then $\lim_{x\rightarrow\infty}\ln\frac{4 + x}{1 + x}=\ln(\lim_{x\rightarrow\infty}\frac{4 + x}{1 + x})=\ln(1)=0$.

Answer:

$0$