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Chapter 2: Problem 6

Interpret \(|f(x)-L|<\varepsilon\) in words.

Short Answer

Expert verified
Answer: The expression \(|f(x) - L| < \varepsilon\) signifies that the function \(f(x)\) gets arbitrarily close to its limit \(L\) as the input \(x\) approaches a certain point. The difference between the function value \(f(x)\) and its limit \(L\) is smaller than the positive value \(\varepsilon\), indicating that the function approaches its limiting value within a small distance.

Step by step solution

01

Identify the components of the expression

In the expression \(|f(x) - L| < \varepsilon\), we have: - \(f(x)\): the value of the function at point \(x\) - \(L\): the limit of the function as \(x\) approaches a certain point - \(| \cdot |\): absolute value function - \(\varepsilon\): a small positive number (called epsilon)
02

Interpret the meaning of absolute value

The absolute value function \(| \cdot |\) measures the distance between two values on the number line. In this case, \(|f(x) - L|\) represents the distance between the value of the function \(f(x)\) and its limit \(L\) as \(x\) approaches a certain point.
03

Interpret the inequality

The inequality \(|f(x) - L| < \varepsilon\) means that the distance between the function value \(f(x)\) and its limit \(L\) is less than \(\varepsilon\). In other words, the difference between \(f(x)\) and \(L\) is smaller than \(\varepsilon\).
04

Express the interpretation in words

In words, the expression \(|f(x) - L| < \varepsilon\) means that the function \(f(x)\) gets arbitrarily close to its limit \(L\) as the input \(x\) approaches a certain point. The value of \(f(x)\) is within the small distance of \(\varepsilon\) from the limit \(L\), emphasizing the closeness the function approaches its limiting value.

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The hyperbolic cosine function, denoted cosh \(x\), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as \(\cosh x=\frac{e^{x}+e^{-x}}{2}\). a. Determine its end behavior by evaluating \(\lim \cosh x\) and \(\lim _{x \rightarrow-\infty} \cosh x\). b. Evaluate cosh 0. Use symmetry and part (a) to sketch a plausible graph for \(y=\cosh x\).

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