13. -/1 points details my notes sesscalcet2 3.7.031.mi. find the limit. use lhospitals rule if appropriate…

Explanation:

Step1: Analyze the two - part function

We know that $\lim_{x\rightarrow\infty}3x=\infty$ and $\lim_{x\rightarrow\infty}\ln x=\infty$. But the growth rate of $y = 3x$ is much faster than $y=\ln x$.

Step2: Consider the difference

As $x$ approaches infinity, $3x$ grows without bound much faster than $\ln x$. So, $\lim_{x\rightarrow\infty}(3x-\ln x)=\lim_{x\rightarrow\infty}x(3 - \frac{\ln x}{x})$. We know that $\lim_{x\rightarrow\infty}\frac{\ln x}{x}$ can be evaluated using L'Hopital's Rule. Differentiating the numerator and denominator, if $y=\frac{\ln x}{x}$, then $y'=\frac{\frac{1}{x}}{1}=\frac{1}{x}$. And $\lim_{x\rightarrow\infty}\frac{1}{x}=0$. So, $\lim_{x\rightarrow\infty}(3x - \ln x)=\lim_{x\rightarrow\infty}x(3 - \frac{\ln x}{x})=\infty$.

Answer:

$\infty$