if f is the function defined by f(x) = (1/x - 1)/(x - 1), then lim f(x) as x→1 is equivalent to which of the…

if f is the function defined by f(x) = (1/x - 1)/(x - 1), then lim f(x) as x→1 is equivalent to which of the following? a lim ( - 1/x) as x→1 b lim (1/x² - 1) as x→1 c lim ((e - 1)/(x - 1)) as x→1 d (lim(1/x - 1) as x→1)/(lim(x - 1) as x→1)

if f is the function defined by f(x) = (1/x - 1)/(x - 1), then lim f(x) as x→1 is equivalent to which of the following? a lim ( - 1/x) as x→1 b lim (1/x² - 1) as x→1 c lim ((e - 1)/(x - 1)) as x→1 d (lim(1/x - 1) as x→1)/(lim(x - 1) as x→1)

Answer

Explanation:

Step1: Simplify the function $f(x)$

First, simplify $f(x)=\frac{\frac{1}{x}-1}{x - 1}=\frac{\frac{1 - x}{x}}{x - 1}=\frac{1 - x}{x(x - 1)}=-\frac{1}{x}$ for $x\neq1$.

Step2: Find the limit

$\lim_{x\rightarrow1}f(x)=\lim_{x\rightarrow1}\frac{\frac{1}{x}-1}{x - 1}=\lim_{x\rightarrow1}-\frac{1}{x}$.

Answer:

A. $\lim_{x\rightarrow1}(-\frac{1}{x})$