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Chapter 2: Problem 12

Estimating a limit from tables Let \(f(x)=\frac{x^{3}-1}{x-1}\) a. Calculate \(f(x)\) for each value of \(x\) in the following table. b. Make a conjecture about the value of \(\lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1}\)

Short Answer

Expert verified
Answer: The conjectured limit is 3.

Step by step solution

01

Calculate f(x) for each value of x in the table

To calculate \(f(x)\) for the values of \(x\) in the table, we will simply plug in the value of \(x\) into the function for each entry and simplify: \(f(x)=\frac{x^3-1}{x-1}\) The table values are not provided, so we will create a small table with some example values for \(x\): | x | f(x) | |---|------| | 0.9| | | 0.99| | | 1.01| | | 1.1 | | Calculating the values of \(f(x)\) for each value of \(x\): \(-\frac{(0.9)^3-1}{0.9-1}=0.9^2+0.9+1\approx2.71\) \(-\frac{(0.99)^3-1}{0.99-1}=0.99^2+0.99+1\approx2.9701\) \(\frac{(1.01)^3-1}{1.01-1}=1.01^2+1.01+1\approx3.0301\) \(\frac{(1.1)^3-1}{1.1-1}=1.1^2+1.1+1\approx3.31\) Now, our table is: | x | f(x) | |---|------| | 0.9| 2.71 | | 0.99| 2.9701| | 1.01| 3.0301| | 1.1 | 3.31 |
02

Make a conjecture about the limit of f(x) as x approaches 1

The table shows that the value of \(f(x)\) gets closer to \(3\) as \(x\) approaches \(1\). We can observe this from the table entries: When \(x\) is slightly less than 1 (\(0.9\) and \(0.99\)), \(f(x)\) is also slightly less than 3 (\(2.71\) and \(2.9701\)). When \(x\) is slightly greater than 1 (\(1.01\) and \(1.1\)), \(f(x)\) is also slightly greater than 3 (\(3.0301\) and \(3.31\)). Given this pattern, it seems likely that the limit of \(f(x)\) as \(x\) approaches 1 is 3. Therefore, our conjecture is: \(\lim_{x \rightarrow 1} \frac{x^3-1}{x-1} = 3\)

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Key Concepts

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

Estimating Limits
In calculus, understanding how to estimate the limit of a function as the variable approaches a particular value is crucial. This involves observing the behavior of the function for values close to the point of interest. For our exercise, we deal with the function \(f(x)=\frac{x^3-1}{x-1}\). By evaluating this function at different values of \(x\) near 1, we can estimate the limit of \(f(x)\) as \(x\) approaches 1.Here's how you do it:
  • Choose values that are very close to 1, both smaller and larger, such as 0.9, 0.99, 1.01, and 1.1.
  • Calculate \(f(x)\) for each of these values.
  • Observe how these results behave as \(x\) gets very close to 1.
By looking at the calculated values, you notice a pattern emerging. They inch closer and closer to a specific number, which in this case is 3. This suggests that the limit of the function as \(x\) approaches 1 is likely 3. This is an empirical approach to limit estimation based on the function's output as the input nears a point of singular interest.
Function Evaluation
Function evaluation is the process of calculating the output of a function given certain values of the variable. In this exercise, we evaluated \(f(x) = \frac{x^3-1}{x-1}\) for values of \(x\) near 1.As you substitute each value into \(f(x)\), you perform the arithmetic operations to simplify the expression. This can sometimes include a bit of algebra, especially when working with polynomial fractions like this one. Let's break it down:
  • Substitute \(x\) with the given number. For example, for \(x=0.9\), you substitute it into the function \(\frac{(0.9)^3-1}{0.9-1}\).
  • Simplify the expression by performing the calculations, ensuring to maintain accuracy with decimals.
  • Repeat for other values such as 0.99, 1.01, and 1.1.
Through evaluation, you discern how the function behaves around the point of interest. It's like peeking into the function's characteristics, one number at a time.
Conjecture Formulation
Formulating a conjecture involves making a reasoned guess based on observed data. In our case, after evaluating \(f(x)\) for values approaching 1, you notice the outputs gravitating towards a consistent number.The steps in arriving at a conjecture are:
  • Observe the calculated values of \(f(x)\) and identify a trend. In this problem, values like 2.71, 2.9701, 3.0301, and 3.31 suggest an approach towards 3.
  • Understand the tendency of the function, whether it increases, decreases, or levels off around the point of interest.
  • Based on these observations, posit that \(\lim_{x \to 1} \frac{x^3-1}{x-1} = 3\).
This conjecture captures the limit behavior inferred from the evaluated values. It shows how analytic predictions can be supported by numerical evidence. By systematically evaluating and observing these patterns, you build a foundation for more advanced mathematical understanding, emphasizing the importance of making and testing conjectures in calculus.

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Most popular questions from this chapter

a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{\sqrt{x^{2}+2 x+6}-3}{x-1}$$

Suppose \(f\) is continuous at \(a\) and assume \(f(a)>0 .\) Show that there is a positive number \(\delta>0\) for which \(f(x)>0\) for all values of \(x\) in \((a-\delta, a+\delta)\). (In other words, \(f\) is positive for all values of \(x\) sufficiently close to \(a .\) )

Use the following definitions. Assume fexists for all \(x\) near a with \(x>\) a. We say that the limit of \(f(x)\) as \(x\) approaches a from the right of a is \(L\) and write \(\lim _{x \rightarrow a^{+}} f(x)=L,\) if for any \(\varepsilon>0\) there exists \(\delta>0\) such that $$ |f(x)-L|<\varepsilon \quad \text { whenever } \quad 00\) there exists \(\delta>0\) such that $$ |f(x)-L|<\varepsilon \quad \text { whenever } \quad 0

Graph the function \(f(x)=\frac{\sin x}{x}\) using a graphing window of \([-\pi, \pi] \times[0,2] .\) a. Sketch a copy of the graph obtained with your graphing device and describe any inaccuracies appearing in the graph. b. Sketch an accurate graph of the function. Is \(f\) continuous at \(0 ?\) c. Conjecture the value \(\lim _{x \rightarrow 0} \frac{\sin x}{x}.\)

Evaluate the following limits. $$\lim _{x \rightarrow \pi / 2} \frac{\sin x-1}{\sqrt{\sin x}-1}$$
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