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Chapter 11: Problem 6

construct a table to find the indicated limit. $$\lim _{x \rightarrow 2}\left(x^{2}-1\right)$$

Short Answer

Expert verified
The limit of the function \(f(x)=x^{2}-1\) as \(x\) approaches 2 is 3.

Step by step solution

01

Create a table

The table should have two columns: one for the \(x\) values and one for the corresponding \(y\) values. The \(x\) values should represent a series of numbers that get arbitrarily close to 2 from both sides. For instance, select \(x\) values as follows: 1.9, 1.99, 1.999, 2.001, 2.01, and 2.1.
02

Calculate corresponding \(y\) values

For each selected \(x\) value, calculate the corresponding \(y\) value using the function \(f(x)=x^{2}-1\). Write down the results in the second column of the table.
03

Approximate the limit

Examine the \(y\) values as \(x\) values get closer and closer to 2. The number that the \(y\) values approach as \(x\) approaches 2 is the limit. You'll notice that as \(x\) gets arbitrarily close to 2 from both sides, the \(y\) values approach 3.

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Key Concepts

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

Utilizing a Table of Values in Calculus
When dealing with limits in calculus, a table of values can be an extremely helpful tool. Essentially, this method involves selecting a series of numbers, called \( x \) values, that gradually get closer to the target value from both directions. In our example problem, the objective is to understand the behavior of the function \( x^2 - 1 \) as \( x \) gets close to 2. To do this effectively:
  • Pick numbers slightly less than 2, like 1.9, 1.99, and 1.999.
  • Pick numbers slightly more than 2, such as 2.001, 2.01, and 2.1.
By placing these into the first column of the table and calculating the corresponding \( y \) values in the second column, we can better visualize how the function behaves as \( x \) nears the limit point.
Exploring Function Evaluation
Function evaluation is the process of determining the output value of a function for specific input numbers. In our problem, the function \( f(x) = x^2 - 1 \) is being evaluated at each of our chosen \( x \) values. By substituting these \( x \) values into the function, we compute the corresponding \( y \) values:
  • For \( x = 1.9 \), \( f(1.9) = 1.9^2 - 1 \).
  • For \( x = 2.1 \), \( f(2.1) = 2.1^2 - 1 \).
This simple act of computation allows us to populate the second column of our table of values, showcasing the function's behavior as the inputs vary. Calculating these values helps us understand the function's response as \( x \) approaches the limit.
Understanding the Concept of Approaching a Limit
In calculus, approaching a limit is about understanding what happens to the outputs of a function when the inputs get infinitely close to a particular point. This is the crux of the limit concept. From our filled table:
  • Notice how the \( y \) values get progressively closer to a single number, as \( x \) gets nearer to 2.
  • By observing, we deduce that the \( y \) values converge towards 3.
This convergence indicates that the limit of \( f(x) = x^2 - 1 \) as \( x \) approaches 2 is indeed 3. Understanding limits in terms of approaching values provides a fundamental insight into the behavior of functions, especially in the context of calculus.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. For the function $$f(x)=\left\\{\begin{array}{ll}x^{2} & \text { if } x<1 \\\A x-3 & \text { if } x \geq 1\end{array}\right.$$ find \(A\) so that the function is continuous at 1.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. \(f\) and \(g\) are both continuous at \(a\), although \(\frac{f}{g}\) is not.

The following piecewise function gives the tax owed, \(T(x)\) by a single taxpayer for a recent year on a taxable income of \(x\) dollars. $$T(x)=\left\\{\begin{array}{cl}0.10 x & \text { if } \quad 0 < x \leq 8500 \\\850.00+0.15(x-8500) & \text { if } \quad 8500 < x \leq 34,500 \\\4750.00+0.25(x-34,500) & \text { if } 34,500 < x \leq 83,600 \\\17,025.00+0.28(x-83,600) & \text { if } 83,600 < x \leq 174,400 \\\42,449.00+0.33(x-174,400) & \text { if } 174,400 < x \leq 379,150 \\\110,016.50+0.35(x-379,150) & \text { if } \quad \quad \quad x > 379,150\end{array}\right.$$ a. Determine whether \(T\) is continuous at 8500 . b. Determine whether \(T\) is continuous at \(34,500\). c. If \(T\) had discontinuities, use one of these discontinuities to describe a situation where it might be advantageous to earn less money in taxable income.

Without showing the details, explain how to use a table to find \(\lim _{x \rightarrow 4} x^{2}\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. If \(\lim _{x \rightarrow a^{-}} f(x) \neq f(a)\) and \(\lim _{x \rightarrow a^{-}} f(x) \neq \lim _{x \rightarrow a^{+}} f(x)\), I can redefine \(f(a)\) to make \(f\) continuous at \(a\).
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