question 17 (1 point) find the limit. lim(x→∞) (√(x² - x))/(3x + 1)

Explanation:

Step1: Divide numerator and denominator by x

As (x\to\infty), (x>0), so (\sqrt{x^{2}} = x). Divide (\frac{\sqrt{x^{2}-x}}{3x + 1}) by (x) (equivalent to dividing numerator by (\sqrt{x^{2}}) and denominator by (x)): (\lim_{x\to\infty}\frac{\sqrt{x^{2}-x}}{3x + 1}=\lim_{x\to\infty}\frac{\sqrt{1-\frac{1}{x}}}{3+\frac{1}{x}})

Step2: Use limit - rules

We know that (\lim_{x\to\infty}\frac{1}{x}=0). By the quotient - rule of limits (\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}) ((\lim_{x\to a}g(x)\neq0)) and the composition - rule of limits. (\lim_{x\to\infty}\sqrt{1 - \frac{1}{x}}=\sqrt{\lim_{x\to\infty}(1-\frac{1}{x})}=1) and (\lim_{x\to\infty}(3+\frac{1}{x}) = 3).

Answer:

(\frac{1}{3})