/*! 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 67 Try it-you just keep on cycling.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 67

Try it-you just keep on cycling. Find the limits some other way. $$\lim _{x \rightarrow \infty} \frac{\sqrt{9 x+1}}{\sqrt{x+1}}$$

Short Answer

Expert verified
The limit is 3.

Step by step solution

01

Write the Limit Expression

We start by writing the given limit expression: \[\lim _{x \rightarrow \infty} \frac{\sqrt{9x+1}}{\sqrt{x+1}}\]Our goal is to find the limit as \(x\) approaches infinity.
02

Simplify the Expression

We'll simplify the expression inside the limit by factoring out \(\sqrt{x}\) from both the numerator and the denominator.The expression becomes:\[\lim _{x \rightarrow \infty} \frac{\sqrt{x}\sqrt{9 + \frac{1}{x}}}{\sqrt{x}\sqrt{1 + \frac{1}{x}}}\] Notice how \(\sqrt{x}\) can be canceled out from the numerator and the denominator.
03

Cancel Common Terms

Now, cancelling \(\sqrt{x}\) from the numerator and the denominator, we get:\[\lim _{x \rightarrow \infty} \frac{\sqrt{9 + \frac{1}{x}}}{\sqrt{1 + \frac{1}{x}}}\] This step reduces the expression significantly, clarifying the behavior of the expression as \(x\) approaches infinity.
04

Evaluate the Limit

Note that as \(x\) goes to infinity, \(\frac{1}{x}\) approaches zero. Therefore, the expression simplifies to:\[\lim _{x \rightarrow \infty} \frac{\sqrt{9 + 0}}{\sqrt{1 + 0}} = \frac{\sqrt{9}}{\sqrt{1}} = \frac{3}{1} = 3\]Hence, the limit is 3.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Simplifying expressions
When dealing with complex mathematical expressions, simplifying them can make finding the solution much clearer. In limits, especially when involving square roots or other intricate functions, simplifying expressions helps reveal underlying patterns or behaviors. For example, in the step-by-step solution provided, simplifying helps us tackle the limit as \(x\) approaches infinity by factoring out common terms. This is a starting point to understanding how the function behaves.
  • Identify similar terms: Recognizing components of the expression that repeat can allow you to factor and simplify.
  • Factor common elements: Taking out common factors, like \(\sqrt{x}\) in the example, often makes these terms easier to cancel and interpret.
  • Cancel and reduce: By cancelling common terms, you streamline the expression, making it much easier to evaluate as \(x\) changes.
Breaking down problems in this way reduces complexity and facilitates a clearer path toward finding a solution.
Infinity in calculus
Infinity in calculus is a fascinating concept since it denotes a number larger than any other conceivable number. It's crucial to understand how functions behave as we approach infinity to find limits effectively. In the context of the exercise, we explore what happens as \(x\) grows indefinitely.
  • Approaching infinity: In many limit problems, as \(x\) approaches infinity, specific terms either disappear (like \(\frac{1}{x}\) approaching zeros) or dominate.
  • Growth rates: Identifying how terms in a function grow relative to one another helps determine how they transform as \(x\) increases.
  • Simplification impact: As variables like \(x\) head towards these extreme values, simplified expressions showcase actual impacts of each term better.
This understanding is very significant when simplifying and solving expressions where infinity plays a central role, such as in this problem.
Factoring in calculus
Factoring, often used in algebra, also plays a pivotal role in calculus. When tackling limits, factoring can simplify the expression and expose relationships between terms. In calculus, it's not only about dealing with polynomial equations but also simplifying under roots and fractions.
  • Identify common factors: Spotting terms that can be factored out or extracted aids in reducing the complexity of a function.
  • Reduce to essentials: By factoring out \(\sqrt{x}\), as done in the solution, the expression becomes more manageable and intuitive.
  • Enhance understanding: This process often leads to discovering how the separate parts of a function behave together as a whole, especially near extremes like infinity.
Factoring is thus a bridge from a complex-looking expression to its simpler, core components, easing the process of evaluating limits and other calculus operations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve the initial value problems in Exercises. Find the curve \(y=f(x)\) in the \(x y\) -plane that passes through the point (9,4) and whose slope at each point is \(3 \sqrt{x}\)

Solve the initial value problems in Exercises. $$\begin{aligned} &y^{(4)}=-\sin t+\cos t\\\ &y^{\prime \prime \prime}(0)=7, \quad y^{\prime \prime}(0)=y^{\prime}(0)=-1, \quad y(0)=0 \end{aligned}$$

Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. How are the values at which these graphs intersect the \(x\)-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function? $$y=x^{5}-5 x^{4}-240$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=\cos ^{-1}\left(x^{2}\right)$$

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$f(x)=\ln (2 x+x \sin x), \quad[1,15]$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.