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

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

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

Answer

Explanation:

Step1: Simplify the function

Given (f(x)=\frac{\frac{1}{x}-1}{x - 1}), first simplify the numerator. (\frac{1}{x}-1=\frac{1 - x}{x}). So (f(x)=\frac{\frac{1 - x}{x}}{x - 1}=\frac{1 - x}{x(x - 1)}). Since (1 - x=-(x - 1)), then (f(x)=\frac{-(x - 1)}{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}) (because the functions (\frac{\frac{1}{x}-1}{x - 1}) and (-\frac{1}{x}) are equal for (x\neq1)).

Answer:

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