Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Problem 690

\(\lim _{\mathrm{x} \rightarrow 2}\left[\left(4-8 \mathrm{x}+5 \mathrm{x}^{2}-\mathrm{x}^{3}\right) /\left(2 \mathrm{x}^{3}-9 \mathrm{x}^{2}+12 \mathrm{x}-4\right)\right]=?\) (a) \((1 / 3)\) (b) \(-(1 / 3)\) (c) 3 (d) \(-3\)

Short Answer

Expert verified
\(-\frac{1}{2}\)

Step by step solution

01

Try direct substitution

Check if the expression is defined at x=2 by plugging in this value. If not, proceed to factorization. \[f(x)=\frac{4-8x+5x^2-x^3}{2x^3-9x^2+12x-4}\] Let's find f(2): \[f(2)=\frac{4-8(2)+5(2)^2-(2)^3}{2(2)^3-9(2)^2+12(2)-4}\] \[f(2)=\frac{4-16+20-8}{16-36+24-4}\] \[f(2)=\frac{0}{0}\] Since we get an indeterminate form (0/0), we can't just substitute x=2. We need to factorize the numerator and denominator to simplify.
02

Factorize the numerator and the denominator

Factorize the polynomials in the numerator and denominator to simplify the expression: The numerator is a cubic polynomial: \(h(x) = x^3 - 5x^2 + 8x - 4\) The denominator is another cubic polynomial: \(g(x) = 2x^3 - 9x^2 + 12x - 4\) By inspection or using polynomial division, we can factor both polynomials: \(h(x) = (x-2)(x^2-3x+2)\) \(g(x) = (x-1)(2x^2-7x+4)\)
03

Simplify the expression and find the limit

After factorizing the polynomials, substitute the factors back into the expression for the limit and simplify by canceling common factors: \[\lim_{x\to 2} \frac{(x-2)(x^2-3x+2)}{(x-1)(2x^2-7x+4)}\] We can cancel the (x-2) factor in the numerator and denominator: \[\lim_{x\to 2} \frac{x^2-3x+2}{(x-1)(2x^2-7x+4)}\] Now, we can directly substitute x=2 into the simplified expression: \[\lim_{x\to 2} \frac{2^2-3(2)+2}{(2-1)(2(2)^2-7(2)+4)}\] \[\lim_{x\to 2} \frac{4-6+2}{(1)(8-14+4)}\] \[\lim_{x\to 2} \frac{0}{(-2)}\] The limit is equal to 0/-2: \[\lim_{x\to 2} \frac{4-8x+5x^2-x^3}{2x^3-9x^2+12x-4}= -\frac{1}{2}\] The correct answer is not among the given choices.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

If \(\mathrm{f}(\mathrm{x})=\left[\left(\mathrm{e}^{(1 / \mathrm{x})}-\mathrm{e}^{-(1 / \mathrm{x})}\right) /\left(\mathrm{e}^{(1 / \mathrm{x})}+\mathrm{e}^{-(1 / \mathrm{x})}\right)\right], \mathrm{x} \neq 0\) and \(\lim _{\mathrm{x} \rightarrow(0+)+} \mathrm{f}(\mathrm{x})=\mathrm{a}, \lim _{\mathrm{x} \rightarrow(0)-} \mathrm{f}(\mathrm{x})=\mathrm{b}\) then the value of \(\mathrm{a}\) and \(\mathrm{b}\) are: (a) \(\mathrm{a}=1, \mathrm{~b}=-1\) (b) \(a=0, b=1\) (c) \(a=-1, b=1\) (d) \(\mathrm{a}=1, \mathrm{~b}=0\)

\(\lim _{\mathrm{x} \rightarrow 1}[\sqrt{\\{1-\operatorname{Cos} 2(\mathrm{x}-1)\\} /(\mathrm{x}-1)}]=?\) (a) \(\sqrt{2}\) (b) 1 (c) Limit does not exist (d) \(-\sqrt{2}\)

\(\lim _{\mathrm{x} \rightarrow 0}\left[(\sin 2 \mathrm{x}-\tan 2 \mathrm{x}) / \mathrm{x}^{3}\right]=?\) (a) 4 (b) \(-8\) (c) \(-4\) (d) 8

If \((\mathrm{a} / 2)\) and \((\mathrm{b} / 2)\) be two distinct real roots of \(\ell \mathrm{x}^{2}+\mathrm{mx}+\mathrm{n}=0\) then \(\lim _{\mathrm{x} \rightarrow(\mathrm{a} / 2)}\left[\left\\{1-\operatorname{Cos}\left(\ell \mathrm{x}^{2}+\mathrm{mx}+\mathrm{n}\right)\right\\} /(2 \mathrm{x}-\mathrm{a})^{2}\right]=?\) (Where \(\ell=0, \mathrm{a}, \mathrm{b} \in \mathrm{R})\) (a) \(\left[\ell^{2} /\left\\{8(\mathrm{a}-\mathrm{b})^{2}\right\\}\right]\) (b) \(\left(\ell^{2} / 32\right)\left(a^{2}-b^{2}\right)\) (c) \(\left(\ell^{2} / 32\right)(\mathrm{a}-\mathrm{b})^{2}\) (d) \(\left(\ell^{2} / 16\right)\left(a^{2}-b^{2}\right)\)

The value \(\mathrm{p}\) for which the function \(\mathrm{f}(\mathrm{x})=\left[\left(4^{\mathrm{x}}-1\right)^{3} /\left\\{\operatorname{Sin}(\mathrm{x} / \mathrm{p}) \log \left[1+\left(\mathrm{x}^{2} / 3\right)\right]\right\\}\right], \mathrm{x} \neq 0\) \(\mathrm{f}(\mathrm{x})=12(\log 4)^{3}, \mathrm{x}=0\) may be continuous at \(\mathrm{x}=0\) is : (a) 1 (b) 2 (c) 3 (d) 4
See all solutions

Recommended explanations on English Textbooks

Essay Prompts

Read Explanation

History of English Language

Read Explanation

Free Response Essay

Read Explanation

Lexis and Semantics

Read Explanation

Global English

Read Explanation

Language Analysis

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.

Sign-up for free