/*! 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 85 Use analytical methods to evalua... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 85

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow 6} \frac{\sqrt[5]{5 x+2}-2}{1 / x-1 / 6}$$

Short Answer

Expert verified
Question: Calculate the limit of the given expression as x approaches 6. Expression: $$\lim_{x \rightarrow 6} \frac{\sqrt[5]{5 x+2}-2}{\frac{1}{x}-\frac{1}{6}}$$ Answer: The limit of the given expression as x approaches 6 is $$-\frac{1}{298}$$.

Step by step solution

01

Simplify the Fraction Individually

Start by noticing that in the denominator of the fraction, we can find a common denominator and simplify the terms: $$\frac{1}{x} - \frac{1}{6} = \frac{6-x}{6x}$$
02

Rationalize the Numerator

To simplify the numerator, we can use the technique of rationalization, which involves multiplying and dividing the expression by its conjugate. The conjugate of partition is $$(\sqrt[5]{5x+2}+2)^4(\sqrt[5]{5x+2}^3+2^4\sqrt[5]{5x+2}^2+2^8\sqrt[5]{5x+2}+2^{12})$$. Multiply both the numerator and denominator by this expression: $$[\frac{\sqrt[5]{5 x+2}-2}{\frac{6-x}{6x}}]\times[\frac{(\sqrt[5]{5x+2}+2)^4(\sqrt[5]{5x+2}^3+2^4\sqrt[5]{5x+2}^2+2^8\sqrt[5]{5x+2}+2^{12})}{(\sqrt[5]{5x+2}+2)^4(\sqrt[5]{5x+2}^3+2^4\sqrt[5]{5x+2}^2+2^8\sqrt[5]{5x+2}+2^{12})}]$$
03

Evaluate the Rationalized Expression

Multiplying the rationalized numerator, we get that the numerator simplifies to: $$(5x+2)-32 = 5x-30$$ Now, the expression becomes: $$\frac{5x-30}{(6-x)(6x)(\sqrt[5]{5x+2}+2)^4(\sqrt[5]{5x+2}^3+2^4\sqrt[5]{5x+2}^2+2^8\sqrt[5]{5x+2}+2^{12})}$$
04

Cancel out common factors and Simplify

Cancel out the factor of \((6-x)\) in the numerator and denominator: $$\frac{-5}{(6x)(\sqrt[5]{5x+2}+2)^4(\sqrt[5]{5x+2}^3+2^4\sqrt[5]{5x+2}^2+2^8\sqrt[5]{5x+2}+2^{12})}$$
05

Calculate the Limit

Now, we can evaluate the limit as x approaches 6 $$\lim_{x \rightarrow 6} \frac{-5}{(6x)(\sqrt[5]{5x+2}+2)^4(\sqrt[5]{5x+2}^3+2^4\sqrt[5]{5x+2}^2+2^8\sqrt[5]{5x+2}+2^{12})}$$ Replace x with 6 $$= \frac{-5}{(36)(\sqrt[5]{32}+2)^4(\sqrt[5]{32}^3+2^4\sqrt[5]{32}^2+2^8\sqrt[5]{32}+2^{12})}$$
06

Calculate the Final Result

Simplify the expression in the denominator: $$= \frac{-5}{(36)(2+2)^4(2^5+2^4\cdot2^3+2^8\cdot2+2^{12})}$$ $$= \frac{-5}{(36)(4)^4(2^5+2^7+2^9+2^{12})}$$ $$= \frac{-5}{(36\cdot256)(2^5(1+2^2+2^4+2^7))}$$ $$= \frac{-1}{2(1+4+16+128)}$$ $$= \frac{-1}{2(149)}$$ $$= \frac{-1}{298}$$ So, the limit of the given expression as x approaches 6 is $$\lim_{x \rightarrow 6} \frac{\sqrt[5]{5 x+2}-2}{\frac{1}{x}-\frac{1}{6}} = -\frac{1}{298}$$

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

The complexity of a computer algorithm is the number of operations or steps the algorithm needs to complete its task assuming there are \(n\) pieces of input (for example, the number of steps needed to put \(n\) numbers in ascending order). Four algorithms for doing the same task have complexities of A: \(n^{3 / 2},\) B: \(n \log _{2} n,\) C: \(n\left(\log _{2} n\right)^{2},\) and \(D: \sqrt{n} \log _{2} n .\) Rank the algorithms in order of increasing efficiency for large values of \(n\) Graph the complexities as they vary with \(n\) and comment on your observations.

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime}(x)=\cos x, F^{\prime}(0)=3, F(\pi)=4$$

Estimate \(f(5.1)\) given that \(f(5)=10\) and \(f^{\prime}(5)=-2\)

Consider the functions \(f(x)=\frac{1}{x^{2 n}+1},\) where \(n\) is a positive integer. a. Show that these functions are even. b. Show that the graphs of these functions intersect at the points \(\left(\pm 1, \frac{1}{2}\right),\) for all positive values of \(n\) c. Show that the inflection points of these functions occur at \(x=\pm \sqrt[2 n]{\frac{2 n-1}{2 n+1}},\) for all positive values of \(n\) d. Use a graphing utility to verify your conclusions. e. Describe how the inflection points and the shape of the graphs change as \(n\) increases.

Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow \pi / 2}(\pi-2 x) \tan x$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.