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Chapter 1: Problem 58

Compute the following limits. $$\lim _{x \rightarrow \infty} \frac{1}{x-8}$$

Short Answer

Expert verified
The limit is 0.

Step by step solution

01

Identify the Limit

Determine what is being asked. Here, the limit to evaluate is \(\lim_{x \rightarrow \infty} \frac{1}{x-8} \)
02

Understand the Behavior of the Function

Observe the behavior of the function as x approaches infinity. The function \( \frac{1}{x-8} \) will have a denominator that grows larger and larger.
03

Apply the Limit Law

Understand that as x becomes very large, \( x - 8 \) still becomes very large. Therefore, \( \frac{1}{x-8} \) gets closer and closer to 0 because the numerator is constant and the denominator increases without bound.
04

Conclusion

Since the value of a fraction where the numerator is constant and the denominator increases without bound approaches 0, \(\lim_{x \rightarrow \infty} \frac{1}{x-8} = 0 \)

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Key Concepts

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

limits of functions
Limits of functions are a foundational concept in calculus. They help us understand the value a function approaches as the input (or x-value) gets closer and closer to a specific point. In the context of the exercise, we're dealing with the limit of a function as x approaches infinity. To find a limit, you look at the behavior of the function as the variable changes. This approach can indicate if the function approaches a particular value, becomes unbounded, or oscillates.
behavior of functions
Understanding the behavior of functions is crucial when working with limits. In the provided exercise, we analyzed the behavior of \(\frac{1}{x-8}\) as x approaches infinity. Here are some bullet points to better understand function behavior:
  • As x gets larger, the term \(x-8\) also grows larger.
  • The numerator, which is 1, remains constant.
  • As the denominator increases without bound, the fraction \(\frac{1}{x-8}\) gets closer to 0.

By breaking down how each part of the function behaves, it becomes easier to understand why \(\frac{1}{x-8}\) approaches 0 as x grows large.
infinite limits
Infinite limits help describe how functions behave as the input grows very large (positive infinity) or very small (negative infinity). To determine the limit of \(\frac{1}{x-8}\) as x approaches infinity, we observe:

The numerator remains a constant 1. The denominator \(x-8\) increases indefinitely. Since 1 divided by an increasingly large number gets closer to 0, we conclude that \(\frac{1}{x-8}\) approaches 0.

When working with infinite limits, always consider:
  • How the numerator and denominator behave separately.
  • Whether the entire fraction grows, shrinks or oscillates.

Infinite limits are an essential concept for analyzing functions that grow without bound, helping us predict their long-term behavior.

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Most popular questions from this chapter

Function The revenue from producing (and selling) \(x\) units of a product is given by \(R(x)=3 x-.01 x^{2}\) dollars. (a) Find the marginal revenue at a production level of \(20 .\) (b) Find the production levels where the revenue is 200 dollars.

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 8} \frac{x^{2}+64}{x-8}$$

Find the slope of the tangent line to the curve \(y=x^{3}+3 x-8\) at \((2,6).\)

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. $$f^{\prime}(1), \text { where } f(x)=\sqrt{1+x^{2}}$$

Let \(P(x)\) be the profit from producing (and selling) \(x\) units of goods. Match each question with the proper solution. Questions A. What is the profit from producing 1000 units of goods? B. At what level of production will the marginal profit be 1000 dollars? C. What is the marginal profit from producing 1000 units of goods? D. For what level of production will the profit be 1000 dollars? Solutions (a) Compute \(P^{\prime}(1000)\) (b) Find a value of \(a\) for which \(P^{\prime}(a)=1000\) (c) Set \(P(x)=1000\) and solve for \(x\) (d) Compute \(P(1000)\)
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