/*! 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 100 Consider \(\lim _{x \rightarrow-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 100

Consider \(\lim _{x \rightarrow-\infty} \frac{3 x}{\sqrt{x^{2}+3}}\). Use the definition of limits at infinity to find values of \(N\) that correspond to (a) \(\varepsilon=0.5\) and (b) \(\varepsilon=0.1\).

Short Answer

Expert verified
Here, \( L \) (the limit as \( x \) approaches \( -\infty \)) is found to be \( -3 \). Then, the value of \( N \) for which \( |f(x) - L| < \varepsilon \) holds for \( x > N \) can be searched for by solving the equation \( |f(x) - (-3)| = \varepsilon \) where \( f(x) = \frac{3}{\sqrt{1+\frac{3}{x^2}}}\) for given \( \varepsilon \) values, (0.5 and 0.1).

Step by step solution

01

Simplify the expression

To simplify the function \( f(x) = \frac{3x}{\sqrt{x^2+3}} \), apply the standard technique of multiplying both the numerator and denominator by \( \frac{1}{\sqrt{x^{2}}}\) for \( x \neq 0\). This gives \( f(x) = \frac{3x \cdot \frac{1}{\sqrt{x^{2}}}}{\sqrt{x^{2}+3} \cdot \frac{1}{\sqrt{x^{2}}}} = \frac{3}{\sqrt{1+\frac{3}{x^2}}}\).
02

Calculate the Limit

Now, when the limit of the simplified function \( f(x) = \frac{3}{\sqrt{1+\frac{3}{x^2}}}\) is calculated as \( x \rightarrow -\infty \), the term \( \frac{3}{x^{2}} \) in the denominator becomes zero and the function reduces to \( -3 \). So, \( \lim_{x \rightarrow-\infty} f(x) = -3\). Thus, \( L = -3\).
03

Find N for given \( \varepsilon \)

To find values of \( N \) corresponding to different \( \varepsilon \), use the formula \( |f(x)-L| = \varepsilon \) when \( x > N \). For \( \varepsilon=0.5 \), the formula becomes \( |f(x) - (-3)| = 0.5 \) and the value of \( N \) which obeys this condition should be determined. Similarly for \( \varepsilon=0.1 \), the formula becomes \( |f(x) - (-3)| = 0.1 \). The calculation may require solving a quadratic equation, which depends on the specific values of \( \varepsilon \).
04

Final Comments

For some values of \( \varepsilon \), finding exact solutions for \( N \) might be very challenging or impossible, and approximate solutions might be needed. Also, it鈥檚 important to note that for \( x \rightarrow -\infty \), the limit only holds for \( x < N \), not \( x > N \), as the direction towards \( -\infty \) gets more negative. Therefore, adjust sign conditions accordingly, when solving for \( N \).

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!

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

A section of highway connecting two hillsides with grades of \(6 \%\) and \(4 \%\) is to be built between two points that are separated by a horizontal distance of 2000 feet (see figure). At the point where the two hillsides come together, there is a 50 -foot difference in elevation. (a) Design a section of highway connecting the hillsides modeled by the function \(f(x)=a x^{3}+b x^{2}+c x+d\) \((-1000 \leq x \leq 1000)\). At the points \(A\) and \(B,\) the slope of the model must match the grade of the hillside. (b) Use a graphing utility to graph the model. (c) Use a graphing utility to graph the derivative of the model. (d) Determine the grade at the steepest part of the transitional section of the highway.

Let \(f\) and \(g\) represent differentiable functions such that \(f^{\prime \prime} \neq 0\) and \(g^{\prime \prime} \neq 0\). Prove that if \(f\) and \(g\) are positive, increasing, and concave upward on the interval \((a, b),\) then \(f g\) is also concave upward on \((a, b)\).

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graph of \(f(x)=1 / x\) is concave downward for \(x<0\) and concave upward for \(x>0\), and thus it has a point of inflection at \(x=0\)

The function \(s(t)\) describes the motion of a particle moving along a line. For each function, (a) find the velocity function of the particle at any time \(t \geq 0\), (b) identify the time interval(s) when the particle is moving in a positive direction, (c) identify the time interval(s) when the particle is moving in a negative direction, and (d) identify the time(s) when the particle changes its direction. $$ s(t)=t^{3}-5 t^{2}+4 t $$

Graph a function on the interval [-2,5] having the given characteristics. Relative minimum at \(x=-1\) Critical number at \(x=0,\) but no extrema Absolute maximum at \(x=2\) Absolute minimum at \(x=5\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.