if (f) is the function defined by (f(x)=\frac{\frac{1}{x}-1}{\frac{1}{x^{2}} - 1}), then (lim_{x\rightarrow1}…

if (f) is the function defined by (f(x)=\frac{\frac{1}{x}-1}{\frac{1}{x^{2}} - 1}), then (lim_{x\rightarrow1}f(x)) is equivalent to which of the following?

if (f) is the function defined by (f(x)=\frac{\frac{1}{x}-1}{\frac{1}{x^{2}} - 1}), then (lim_{x\rightarrow1}f(x)) is equivalent to which of the following?

Answer

Explanation:

Step1: Simplify the function

First, simplify (f(x)=\frac{\frac{1}{x}-1}{\frac{1}{x^{2}} - 1}). We know that (\frac{1}{x}-1=\frac{1 - x}{x}) and (\frac{1}{x^{2}}-1=\frac{1 - x^{2}}{x^{2}}=\frac{(1 - x)(1 + x)}{x^{2}}). So (f(x)=\frac{\frac{1 - x}{x}}{\frac{(1 - x)(1 + x)}{x^{2}}}). When (x\neq1), we can cancel out the factor ((1 - x)) (since we are finding the limit as (x\rightarrow1), not evaluating at (x = 1)). Then (f(x)=\frac{x}{1 + x}).

Step2: Find the limit

Now, find (\lim_{x\rightarrow1}f(x)=\lim_{x\rightarrow1}\frac{x}{1 + x}). Substitute (x = 1) into (\frac{x}{1 + x}), we get (\frac{1}{1+1}=\frac{1}{2}).

Answer:

(\frac{1}{2})