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Chapter 2: Problem 6

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

Short Answer

Expert verified
In your own words, explain the meaning of the expression \(|f(x)-L|<\varepsilon\). Answer: The expression \(|f(x)-L|<\varepsilon\) means that the absolute difference between the value of a function, \(f(x)\), and the limit point, \(L\), is smaller than a given positive error tolerance, \(\varepsilon\). In other words, the function's value is very close to the limit within the acceptable range determined by \(\varepsilon\).

Step by step solution

01

Breaking down the expression

We have the expression \(|f(x)-L|<\varepsilon\). Let's break it down into its components: 1. \(f(x)\): This represents a function, which maps an input x to an output f(x). 2. \(L\): Often, when discussing limits, \(L\) represents the limit point, meaning the value that a function converges to as its input x approaches a certain value. 3. \(\varepsilon\): This is a positive number, often referred to as the error tolerance or precision. It represents the maximum difference allowed between the function's value and the limit for the inequality to hold.
02

Interpreting the inequality in words

Now, we can interpret the given inequality \(|f(x)-L|<\varepsilon\) in words: The absolute difference between the function value, \(f(x)\), and the limit point, \(L\), is less than a specified positive error tolerance, \(\varepsilon\).

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Determine whether the following statements are true and give an explanation or counterexample. Assume \(a\) and \(L\) are finite numbers. a. If \(\lim _{x \rightarrow a} f(x)=L,\) then \(f(a)=L\).0 b. If \(\lim _{x \rightarrow a^{-}} f(x)=L,\) then \(\lim _{x \rightarrow a^{+}} f(x)=L\). c. If \(\lim _{x \rightarrow a} f(x)=L\) and \(\lim _{x \rightarrow a} g(x)=L,\) then \(f(a)=g(a)\). d. The limit \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) does not exist if \(g(a)=0\). e. If \(\lim _{x \rightarrow 1^{+}} \sqrt{f(x)}=\sqrt{\lim _{x \rightarrow 1^{+}} f(x)}\), it follows that \(\lim _{x \rightarrow 1} \sqrt{f(x)}=\sqrt{\lim _{x \rightarrow 1} f(x)}\).

Prove the following statements to establish the fact that \(\lim f(x)=L\) if and only if \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\). a. If \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L,\) then \(\lim _{x \rightarrow a} f(x)=L\) b. If \(\lim _{x \rightarrow a} f(x)=L,\) then \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\)
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