/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 71 Let \(f(x)=\left(x^{2}-1\right) ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 71

Let \(f(x)=\left(x^{2}-1\right) /(|x|-1)\) a. Make tables of the values of \(f\) at values of \(x\) that approach \(c=-1\) from above and below. Then estimate \(\lim _{x \rightarrow-1} f(x)\) b. Support your conclusion in part (a) by graphing \(f\) near \(c=-1\) and using Zoom and Trace to estimate \(y\) -values on the graph as \(x \rightarrow-1\) c. Find \(\lim _{x \rightarrow-1} f(x)\) algebraically.

Short Answer

Expert verified
The limit of \(f(x)\) as \(x\) approaches -1 is 2.

Step by step solution

01

Understand the problem

We are given a function \(f(x)=\frac{x^{2}-1}{|x|-1}\). We need to investigate the behavior of this function as \(x\) approaches -1, both from the left (below) and from the right (above). We will do this by creating a table of values near \(x = -1\), graphing the function, and then finding the limit algebraically.
02

Make a table of values approaching c = -1

Calculate the values of \(f(x)\) by choosing values of \(x\) approaching -1 from both below and above.For values approaching from above:- \(x = -0.9\), \(f(x) = \frac{(-0.9)^2 - 1}{|-0.9| - 1} = \frac{0.81 - 1}{0.9 - 1} = \frac{-0.19}{-0.1} = 1.9\)- \(x = -0.99\), \(f(x) = \frac{(-0.99)^2 - 1}{|-0.99| - 1} = \frac{0.9801 - 1}{0.99 - 1} = \frac{-0.0199}{-0.01} = 1.99\)For values approaching from below:- \(x = -1.1\), \(f(x) = \frac{(-1.1)^2 - 1}{|-1.1| - 1} = \frac{1.21 - 1}{1.1 - 1} = \frac{0.21}{0.1} = 2.1\)- \(x = -1.01\), \(f(x) = \frac{(-1.01)^2 - 1}{|-1.01| - 1} = \frac{1.0201 - 1}{1.01 - 1} = \frac{0.0201}{0.01} = 2.01\)
03

Estimate limit from the table

From the table, we can see that as \(x\) approaches -1 from either side, \(f(x)\) approaches 2. Therefore, the estimate for \(\lim_{x \to -1} f(x)\) is 2.
04

Graph the function near c = -1

Plot the function \(f(x)=\frac{x^{2}-1}{|x|-1}\) using a graphing tool focusing on the region near \(x = -1\). Use zoom and trace features to observe how the function behaves as \(x\) approaches -1. The graph should demonstrate that the values are clustering around y = 2 when \(x\) approaches -1.
05

Verify the limit algebraically

Rewrite the function \(f(x)\). For \(x > 0\), \(|x| = x\) and for \(x < 0\), \(|x| = -x\). Thus:- If \(x > 0\), then \(f(x) = \frac{x^2 - 1}{x - 1}\). This simplifies to \(x + 1\).- If \(x < 0\), then \(f(x) = \frac{x^2 - 1}{-x - 1}\). This simplifies to: \(-(x + 1)) = -x - 1\).This means for very small values of \(x\) near -1, \(f(x) = x + 1\) which in both cases tends to 2 as \(x\) tends to -1.Therefore, \(\lim_{x \to -1} f(x) = 2\).
06

Conclusion

By the table, graphical evidence, and algebraic manipulation, the limit of \(f(x)\) as \(x\) approaches -1 is 2.

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Key Concepts

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

Graphical Estimation
Graphical estimation is a useful way to understand the behavior of functions as they approach certain x-values. By plotting a function on a graph, we can visually see how the function behaves when x is near a particular point, such as approaching from the left or right. In this exercise, you can plot the function \( f(x) = \left(\frac{x^2 - 1}{|x| - 1}\right) \) around \( x = -1 \).
  • Plotting the Function: Use graphing software or a calculator to draw the function. Focus near \( x = -1 \)
  • Zoom and Trace: Zoom in on the graph around \( x = -1 \) and use trace tools to move along the curve.
  • Observing Patterns: Notice how \( f(x) \) seems to approach 2 as \( x \) goes closer to -1.
By using these graphical methods, you can visually estimate that the limit of \( f(x) \) as \( x \) approaches -1 is indeed around 2. This method provides a visual confirmation and helps ensure that the table calculations and algebraic manipulations align with graphical evidence.
Algebraic Manipulation
Algebraic manipulation is a crucial skill for finding limits without relying solely on numeric or graphical methods. It involves rearranging and simplifying expressions to understand their behavior at challenging points, such as where division by zero might occur. In our function \( f(x) = \frac{x^2 - 1}{|x| - 1} \), we must address what happens when \(|x|-1 = 0\) as \( x \) approaches -1.
  • Simplification: By recognizing that \( x^2 - 1 = (x - 1)(x + 1) \) using difference of squares, you can attempt simplification depending on the sign of \( x \).
  • Case Analysis: for \( x > 0 \), \(|x| = x \); for \( x < 0 \), \(|x| = -x \), giving affine forms:
    • \( f(x) = x + 1 \) simplifies for \( x eq 1 \) when \( x > 0 \)
    • For negative \( x \), math remains approachable as \( f(x) \rightarrow -(x+1) \)
  • Finding the Limit: Using the these affine expressions, \( f(x) = x + 1 \) means, directly approaching -1, the value \( -1 + 1 = 0 \).
This shows how algebraic manipulation confirms the apparent limit found graphically.
Piecewise Functions
Piecewise functions are functions defined by different expressions over different intervals of their domain. This kind of approach is highly useful when a function behaves differently based on which side of a certain point you are examining it from. For the function \( f(x) = \frac{x^2 - 1}{|x| - 1} \), it effectively acts like a piecewise function as \( x \) can take positive and negative values, which affects the expression for \( |x| \).
  • Understanding Piecewise Elements:
    • Consider \( |x| \) yielding different expressions based on being \( "> " 0\) or \( "<" 0\)
  • Creating the Piecewise Function: By understanding the role of \(|x|\), you see the separate scenarios:
    • \( f(x) = \frac{x^2 - 1}{x-1} \) for \( xeq 1, x > 0 \)
    • \( f(x) = \frac{x^2 - 1}{-x - 1} \) applied for \( x<0 \)
Notice how these definitions guide each algebraic reduction and ultimately calculate the limit in challenging areas. Piecewise descriptions highlight how varying domains provide a broader understanding.

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Most popular questions from this chapter

Here is the definition of infinite right-hand limit. We say that \(f(x)\) approaches infinity as \(x\) approaches \(c\) from the right, and write $$\lim _{x \rightarrow c^{+}} f(x)=\infty$$ if, for every positive real number \(B\), there exists a corresponding number \(\delta>0\) such that for all \(x\) $$cB$$ Modify the definition to cover the following cases. a. \(\lim _{x \rightarrow c^{-}} f(x)=\infty\) b. \(\lim _{x \rightarrow c^{+}} f(x)=-\infty\) c. \(\lim _{x \rightarrow c^{-}} f(x)=-\infty\)

Find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) $$\begin{aligned} &\lim _{x \rightarrow-\infty} h(x)=-1, \lim _{x \rightarrow \infty} h(x)=1, \lim _{x \rightarrow 0^{-}} h(x)=-1, \text { and }\\\ &\lim _{x \rightarrow 0^{+}} h(x)=1 \end{aligned}$$

For what values of \(a\) is $$ f(x)=\left\\{\begin{array}{ll} a^{2} x-2 a, & x \geq 2 \\ 12, & x<2 \end{array}\right. $$ continuous at every \(x ?\)

Sketch the graph of a function \(y=f(x)\) that satisfies the given conditions. No formulas are required-just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) $$\begin{aligned} &f(0)=0, \lim _{x \rightarrow \pm \infty} f(x)=0, \lim _{x \rightarrow 0^{+}} f(x)=2, \text { and }\\\ &\lim _{x \rightarrow 0^{-}} f(x)=-2 \end{aligned}$$

Find the limits \(\lim \frac{x^{2}-3 x+2}{x^{3}-4 x}\) as a. \(x \rightarrow 2^{+}\) b. \(x \rightarrow-2^{+}\) c. \(x \rightarrow 0\) d. \(x \rightarrow 1^{+}\) e. What, if anything, can be said about the limit as \(x \rightarrow 0 ?\)
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