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Chapter 2: Problem 7

Suppose \(|f(x)-5|<0.1\) whenever \(00\) such that \(|f(x)-5|<0.1\) whenever \(0<|x-2|<\delta\).

Short Answer

Expert verified
Answer: The value of 未 that satisfies the desired condition is 未 = 2.

Step by step solution

01

Understand the given information

We are given that \(|f(x) - 5| < 0.1\) whenever \(0 < x < 5\). That means the function \(f(x)\) is always within \(0.1\) units from the value \(5\) in the interval from \(0\) to \(5\). We can represent this as $$4.9 < f(x) < 5.1$$ for every \(x\) in the interval \((0,5)\).
02

Understand the conditions

We want to find possible values for \(\delta\) so that \(|f(x) - 5| < 0.1\) whenever \(0 < |x - 2| < \delta\). This means that, in the interval \((2-\delta, 2+\delta)\), the function \(f(x)\) should be within \(0.1\) units from the value \(5\).
03

Find the intersection between the given and desired conditions

In order to ensure that \(|f(x) - 5| < 0.1\) for \(0 < |x - 2| < \delta\), we need the interval \((2-\delta, 2+\delta)\) to be contained within the interval \((0, 5)\). The intersection of these two intervals will help us determine possible values for \(\delta\).
04

Determine the intersection

The intersection of \((0, 5)\) and \((2-\delta, 2+\delta)\) can be found by considering upper and lower bounds. Both intervals have the following relationship: $$(2-\delta, 2+\delta) \subset (0, 5)$$ The smallest possible value for \(\delta\) will occur when the left endpoints of each interval coincide, while the largest possible value for \(\delta\) will occur when the right endpoints are equal. This gives us the following system of inequalities: $$2 - \delta = 0$$ $$2 + \delta = 5$$
05

Solve the inequalities

Solving the inequalities, we get: $$\delta = 2$$ for the first inequality, and $$\delta = 3$$ for the second inequality.
06

Find all possible values for delta

The inequalities indicate that \(\delta\) should be in the range \((2, 3)\). However, since \(\delta > 0\), we pick the smallest possible value, giving us only one value for \(\delta\): $$\delta=2$$
07

Conclusion

The value of \(\delta\) that satisfies the desired condition is \(\delta = 2\).

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Most popular questions from this chapter

Graph the function \(f(x)=\frac{\sin x}{x}\) using a graphing window of \([-\pi, \pi] \times[0,2] .\) a. Sketch a copy of the graph obtained with your graphing device and describe any inaccuracies appearing in the graph. b. Sketch an accurate graph of the function. Is \(f\) continuous at \(0 ?\) c. Conjecture the value \(\lim _{x \rightarrow 0} \frac{\sin x}{x}.\)

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