Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.7.2

Chapter 2, Problem 2.7.2

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

Verified step by step guidance

Step 1: Understand the problem. We are given that f(x) is in the interval (2, 6), which means 2 < f(x) < 6. We need to find the smallest value of $ \varepsilon $ such that $ |f(x) - 4| < \varepsilon $.

Step 2: Consider the expression $ |f(x) - 4| $. This represents the distance between f(x) and 4 on the number line.

Step 3: Since f(x) is between 2 and 6, the distance from 4 to the nearest endpoint of the interval (2, 6) will determine the smallest $ \varepsilon $.

Step 4: Calculate the distance from 4 to the endpoints of the interval. The distance from 4 to 2 is $ |4 - 2| = 2 $, and the distance from 4 to 6 is $ |6 - 4| = 2 $.

Step 5: The smallest $ \varepsilon $ is the minimum of these distances, which is 2. Therefore, $ \varepsilon = 2 $ ensures that $ |f(x) - 4| < \varepsilon $ for all f(x) in the interval (2, 6).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Inequality

The expression |f(x) - 4| < ε represents an absolute value inequality, which measures the distance between f(x) and the number 4. This inequality states that the value of f(x) must be within ε units of 4, meaning f(x) can vary but must remain close to this central value.

Interval Notation

The interval (2, 6) indicates that the function f(x) takes values strictly between 2 and 6. Understanding this interval is crucial because it helps determine the possible values of f(x) and how they relate to the target value of 4, which is central to the absolute value inequality.

Finding ε

To find the smallest value of ε such that |f(x) - 4| < ε, we need to consider the maximum deviation of f(x) from 4 within the given interval. Since f(x) lies between 2 and 6, the closest points to 4 are 2 and 6, leading to the calculation of ε as the minimum distance from 4 to these endpoints, which is 2.

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Related Practice

Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {√x if x<4

3 if x=4; a=4

x+1 if x>4

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Textbook Question

Determine $\lim_{x \to \infty} f (x)$ and $\lim_{x \to - \infty} f (x)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{4 x}{20 x + 1}$

Textbook Question

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = \frac{3 x^{3} - 7}{x^{4} + 5 x^{2}}$

Textbook Question

Determine the following limits.

lim x→3 1/ x − 3(1 /√x + 1 − 1/2)

Textbook Question

Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

Textbook Question

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?

f(z)=(z−1)^3/4