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Chapter 1: Problem 39

In your own words, describe the meaning of an infinite limit. Is \(\infty\) a real number?

Short Answer

Expert verified
An infinite limit is the value a function tends to as its input nears a certain point; it can be positive or negative infinity. Infinity itself, denoted \( \infty \), is not a real number, but a concept describing an unbounded quantity greater than all real numbers.

Step by step solution

01

Define Infinite Limit

An infinite limit is the value that a function 'f(x)' tends towards as 'x' approaches a certain value. Importantly, this value can be an actual number, positive infinity, or negative infinity, denoted as \( +\infty \) or \( -\infty \) respectively. In other words, a function has an infinite limit if its value becomes arbitrarily large (either positively or negatively) as we approach a certain input value.
02

Discuss the nature of infinity

Infinity, denoted by the symbol \( \infty \), is not a real number, nor is it a number at all. Instead, it serves as a concept to describe something that is endlessly large or unbounded in mathematics. When we say \( \infty \), it simply represents an unbounded quantity that's bigger than any real number.
03

Infinity and Real numbers

Real numbers include both rational and irrational numbers, which can be represented on the number line and can have finite values. Since infinity is not a finite quantity and cannot be represented on the number line, it is not considered a real number.

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Most popular questions from this chapter

In Exercises \(35-38\), use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l} f(x)=\frac{x^{2}+x+1}{x^{3}-1} \\ \lim _{x \rightarrow 1^{+}} f(x) \end{array} $$

(a) Prove that if \(\lim _{x \rightarrow c}|f(x)|=0,\) then \(\lim _{x \rightarrow c} f(x)=0\). (Note: This is the converse of Exercise \(74 .)\) (b) Prove that if \(\lim _{x \rightarrow c} f(x)=L,\) then \(\lim _{x \rightarrow c}|f(x)|=|L|\). [Hint: Use the inequality \(\|f(x)|-| L\| \leq|f(x)-L| .]\)

What is meant by an indeterminate form?

Prove that a function has an inverse function if and only if it is one-to-one

If the functions \(f\) and \(g\) are continuous for all real \(x\), is \(f+g\) always continuous for all real \(x ?\) Is \(f / g\) always continuous for all real \(x ?\) If either is not continuous, give an example to verify your conclusion.
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