/*! 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 22 Determine each limit, if it exis... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 22

Determine each limit, if it exists. $$\lim _{x \rightarrow 0} \frac{x^{2}+2 x}{x}$$

Short Answer

Expert verified
The limit is 2.

Step by step solution

01

Simplify the expression

The given expression is \( \frac{x^2 + 2x}{x} \). We can simplify this expression by dividing the numerator by \( x \), assuming \( x eq 0 \). This gives \( \frac{x(x + 2)}{x} \). The \( x \) in the numerator and denominator cancel out, resulting in \( x + 2 \).
02

Evaluate the limit of the simplified expression

Now, substitute \( x = 0 \) into the simplified expression obtained in Step 1, which is \( x + 2 \). By substituting \( x = 0 \), we get \( 0 + 2 = 2 \).
03

Confirm the existence of the limit

Since we were able to substitute \( x = 0 \) directly after simplifying and found a finite value, the limit exists. The limit is independent of the direction from which \( x \to 0 \) is approached, confirming the existence of the limit.

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
Simplifying expressions is a crucial step when working with limits, especially when you need to handle a function that seems more complicated than it needs to be. In this exercise, the original expression was \( \frac{x^2 + 2x}{x} \). The key to simplifying it was recognizing the common factor of \( x \) in the numerator.
  • Factor the numerator: Start by rewriting \( x^2 + 2x \) as \( x(x + 2) \).
  • Cancel common factors: By dividing both the numerator and the denominator by \( x \), you cancel out the \( x \), given that \( x eq 0 \). This leads you to the simplified form: \( x + 2 \).
Simplification is essential because it often makes evaluating the limit easier. Remember, the domain of the function may exclude the point you are interested in (here \( x eq 0 \)), but simplification can remove complexities to help you see the solution clearly.
Evaluating Limits
Once you've simplified the expression, evaluating the limit becomes much more straightforward. In the given problem, after simplification, the expression was reduced to \( x + 2 \). Evaluating the limit involves substituting the value that \( x \) is approaching, and observing the behavior of the expression.
  • Substitute directly: If the simplified expression isn鈥檛 undefined or indeterminate when you plug in the limit, you can substitute directly. For this problem, substitute \( x = 0 \) into \( x + 2 \).
  • Calculate: The substitution yields \( 0 + 2 \), resulting in \( 2 \).
Direct substitution works perfectly here because the expression is continuous and doesn鈥檛 create an undefined situation at \( x = 0 \). This makes evaluation simple and demonstrates the power of prior simplification.
Existence of Limits
It鈥檚 not enough to simply calculate a value for a limit; you must also examine whether the limit exists. A limit exists if the function approaches a specific value from both directions, as \( x \) approaches the point of interest. Here, after simplifying, you get \( x + 2 \), which is a continuous function at \( x = 0 \).
  • Continuity check: Since \( x + 2 \) doesn鈥檛 have any discontinuities at \( x = 0 \), the substitution confirms the function approaches \( 2 \) from both directions.
  • Direction independence: The limit consistent regardless of approaching from the left or the right of \( x = 0 \), further confirming the limit鈥檚 existence.
In conclusion, establishing the existence of a limit ensures that the solution isn鈥檛 just numerically correct, but also mathematically sound, confirming \( \lim_{x \to 0} (x + 2) = 2 \) holds true.

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 each problem. Evans Price Adjustment Model If there is excess demand for a commodity, the price will rise rapidly at first and then more slowly, according to what economists call the "Evans price adjustment model." A typical function describing this behavior is given by $$ p(t)=12-4 e^{-0.5 t} $$ where \(p(t)\) is the price in dollars after \(t\) days. Calculate \(\lim _{t \rightarrow \infty} p(t)\) and give an interpretation of its value.

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 3} f(x),\) where \(f(x)=\left\\{\begin{array}{ll}x+7 & \text { if } x \leq 3 \\ 5 x-5 & \text { if } x>3\end{array}\right.\)

Solve each problem. Use a calculator to answer each of the following. (a) From a graph of \(y=\frac{\ln x}{x},\) what do you think is the value of \(\lim _{x \rightarrow x} \frac{\ln x}{x} ?\) Support your answer by evaluating the function for several large values of \(x\). (b) Repeat part (a), but this time use the graph of the function \(y=\frac{(\ln x)^{2}}{x}\) (c) On the basis of your results from parts (a) and (b), what do you think is the value of \(\lim _{x \rightarrow \infty} \frac{(\ln x)^{x}}{x},\) where \(n\) is a positive integer?

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{2 x}{\tan x}\)

For the given \(f(x)\), find a formula for \(f^{\prime}(a)\). $$f(x)=x^{2}-5 x$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.