/*! 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 55 find the limit $$ \lim _{\De... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 55

find the limit $$ \lim _{\Delta x \rightarrow 0} \frac{2(x+\Delta x)-2 x}{\Delta x} $$

Short Answer

Expert verified
The value of the limit is 2.

Step by step solution

01

Simplify the Numerator

Attempt to simplify the expression by simplifying its numerator. Begin by simplifying \(2(x+\Delta x)-2 x\). Multiply out the brackets to give \(2x+2\Delta x-2x\). This simplifies to \(2\Delta x\). Our limit therefore simplifies to: \( \lim _{\Delta x \rightarrow 0} \frac{2\Delta x}{\Delta x} \)
02

Cancel Common Terms

Note that \( \Delta x \) appears in both the numerator and denominator. Cancel \( \Delta x \) from both the numerator and the denominator. Our limit now becomes \( \lim _{\Delta x \rightarrow 0} 2 \)
03

Compute the Limit

Now that the indeterminate form has been removed, the limit can be computed directly. However, this limit is simply 2 since 2 is a constant and the limit of a constant is simply the constant itself. Therefore, \( \lim _{\Delta x \rightarrow 0} 2 = 2 \)

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 faced with finding the limit of a function as a variable approaches a certain value, simplifying the expression is crucial to make the problem more manageable.
Simplification includes expanding brackets, combining like terms, and breaking down complex fractions. In our exercise, the process began with expanding the expression within the brackets, resulting in terms that included both the initial variable (\(x\)) and the increment or change (\( \(Delta x \) \) ). By combining like terms, we're often left with a much less cluttered, more straightforward expression, facilitating the application of limit laws and rules.
Canceling Common Terms
Upon simplification, we often encounter expressions where the same term appears in both the numerator and the denominator. These are 'common terms,' which can be canceled out for further simplification.
In the context of limits, canceling out common terms is critical, especially when these terms contain the variable that approaches a certain value. The presence of this variable often signifies an indeterminate form, which canceling can resolve. In our exercise, '\( \(Delta x \) \) ' was canceled from both the numerator and the denominator, streamlining the expression and bringing us a step closer to finding the limit.
Indeterminate Forms
Indeterminate forms are expressions in calculus that do not directly reveal the limit and can take on different values depending on variables' behavior. Common examples include 0/0 and 鈭�/鈭�.
In this exercise, before simplification and canceling, it appears that as '\( \(Delta x \) \) ' approaches zero, our expression would have been undefined, hinting at an indeterminate form. However, simplification and canceling helped us avoid this pitfall by removing the variable, leaving us with a constant. Since the limit of a constant is just the constant itself, resolving the indeterminate form showed that the limit is indeed 2.

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

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\frac{1}{\sqrt{x+2}} $$

Find the point(s), if any, at which the graph of \(f\) has a horizontal tangent. $$ f(x)=\frac{x^{4}}{x^{3}+1} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=\sqrt{4 x^{2}-7} $$

Find \(d y / d u, d u / d x\), and \(d y / d x\). $$ y=u^{2}, u=4 x+7 $$

Find the point(s), if any, at which the graph of \(f\) has a horizontal tangent. $$ f(x)=\frac{x^{2}}{x^{2}+1} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Calculus

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.