\(|[x /(x+1)]|<10^{-5}\) hold if (a) \(-10^{-5}<\mathrm{x}+1<10^{-4}\) (b) \(-(100001)^{-1}<\mathrm{x}<(99999)^{-1}\) (c) \((1 / 10000)<\mathrm{x}<1\) (d) \((99999)^{-1}<\mathrm{x}<(100001)^{-1}\)

In this step, we'll rewrite the inequality \(|x /(x+1)|<10^{-5}\) without the absolute value. Given the quantity \(|x /(x+1)|\), we have two possible cases: the fraction is positive or it is negative. Rewrite the inequality for both cases: Case 1 (Positive): \(0 \le x/(x+1) < 10^{-5}\). Case 2 (Negative): \(-10^{-5} < x/(x+1) \le 0\). Now, we'll solve these inequalities separately.

We are given the inequality \(0 \le x/(x+1)<10^{-5}\). We can first eliminate the \(x = 0\) scenario since this falls under our condition, and we're left with: \(0 \frac{10^{-5}}{1-10^{-5}}\) which simplifies to: \(x > \frac{1}{99999}\)

We now consider the inequality \(-10^{-5} < x/(x+1) \le 0\). We can eliminate the inequality at 0, and we're left with: \(-10^{-5} -10^{-5}\) Now divide by \(1-10^{-5}\): \(x < \frac{-10^{-5}}{1-10^{-5}}\), which simplifies to: \(x < \frac{-1}{100001}\)

We found that \(x\) needs to be in one of the following intervals: \(x > \frac{1}{99999}\) or \(x < \frac{-1}{100001}\) Now let's compare these intervals to the given options: (a) \(-10^{-5}<\mathrm{x}+1<10^{-4}\): This is incorrect, as it doesn't match any of the above intervals we found. (b) \(-(100001)^{-1}<\mathrm{x}<(99999)^{-1}\): This interval matches the intervals we found, with \(x > \frac{1}{99999}\) and \(x < \frac{-1}{100001}\), so this option holds true. (c) \((1 / 10000)<\mathrm{x}<1\): This interval doesn't correspond with the intervals we found. (d) \((99999)^{-1}<\mathrm{x}<(100001)^{-1}\): This interval only matches the positive part of the intervals, neglecting the negative part, so it is partially correct, but option (b) is more accurate. From our analysis, the correct answer is option (b).