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Chapter 1: Problem 6

Find the limit. $$ \lim _{x \rightarrow-2} x^{3} $$

Short Answer

Expert verified
Therefore, the limit of the function \(x^3\) as \(x\) approaches -2 is -8.

Step by step solution

01

Understand the limit notation

The notation \(\lim _{x \rightarrow-2} x^{3}\) is read as 'the limit of \(x^3\) as \(x\) approaches -2.'
02

Substitute the value into the function

To find the limit, we substitute \(x = -2\) into the function \(x^3\). This gives us (-2)^3
03

Evaluate (-2) cubed

The cube of a negative number is negative. Specifically, (-2)^3 equals -8

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit Notation
Limit notation is the mathematical way of expressing the behavior of a function as its input approaches a certain value. In calculus, this concept is pivotal for understanding how functions behave near specific points, even points where the function might not be explicitly defined. For instance, the notation \( \lim _{x \rightarrow-2} x^{3} \) signifies that we want to find out what the cubic function \(x^3\) approaches as \(x\) gets closer and closer to -2. Understanding this notation is the first step in dealing with limits, as it sets up the problem for further analysis.
Substituting Limits
Once you're acquainted with the limit notation, the next step is to substitute the limit value into the function if possible. This technique, known as direct substitution, is the simplest way to evaluate limits when the function is continuous at the point approaching the limit. In the exercise, \(x\) approaches -2, and we're dealing with the function \(x^3\). Substituting -2 in place of \(x\), we get \( (-2)^3 \). It's important to handle substitution carefully, ensuring to preserve the order of operations and the signs to avoid errors in evaluation.
Evaluating Functions
Evaluating functions involves calculating the exact value of a function given a particular input. In the context of limits, once we substitute the approaching value into the function, we evaluate the expression to find the limit. For the cubic function given in the exercise, evaluating \( (-2)^3 \) requires us to cube -2. An understanding that cubing a negative number yields a negative result is essential. Thus, \( (-2)^3 \) gives -8, which is the limit of the function \(x^3\) as \(x\) approaches -2. Such evaluations are the culmination of understanding and applying limit concepts, and they are vital for solving more complex problems in calculus.

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Most popular questions from this chapter

Discuss the continuity of the function on the closed interval. $$ g(x)=\sqrt{25-x^{2}} \quad[-5,5] $$

Discuss the continuity of each function. $$ f(x)=\frac{1}{2} \pi x \rrbracket+x $$

Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 2} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} x^{2}-4 x+6, & x<2 \\ -x^{2}+4 x-2, & x \geq 2 \end{array}\right. $$

Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\frac{x}{x^{2}-x} $$

Find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? $$ f(x)=\frac{x-3}{x^{2}-9} $$
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