Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Problem 718

For the function $$ \mathrm{f}(\mathrm{x})=\mid \begin{aligned} &{\left[\left(2^{-\mathrm{m}}-\mathrm{x}^{\mathrm{m}}\right) /\left(\mathrm{x}^{-\mathrm{m}}-2^{\mathrm{m}}\right)\right]: \mathrm{x} \neq 0.5} \\ &+0.0625 \end{aligned} $$ If \(\mathrm{f}\) is continuous at \(\mathrm{x}=0.5\) then the value of \(\mathrm{m}=\ldots \ldots\) (a) \(0.5\) (b) 2 (c) \(-2\) (d) \(-0.5\)

Short Answer

Expert verified
The value of \(m\) for which the function is continuous at \(x=0.5\) is \(m=2\).

Step by step solution

01

Find the left-hand limit

To find the left-hand limit, we will look at the function as x approaches 0.5 from the left side (x < 0.5). $$\lim_{x \to 0.5^-} \frac{(2^{-m}-x^m)}{(x^{-m}-2^m)}$$
02

Find the right-hand limit

To find the right-hand limit, we will look at the function as x approaches 0.5 from the right side (x > 0.5). $$\lim_{x \to 0.5^+} \frac{(2^{-m}-x^m)}{(x^{-m}-2^m)}$$
03

Check if the left-hand limit equals the right-hand limit

We will examine if the left-hand limit as x approaches 0.5 is equal to the right-hand limit as x approaches 0.5. Note that both the left-hand and right-hand limits are of the same form, so we'll rewrite the limit as: $$\lim_{x \to 0.5} \frac{(2^{-m}-x^m)}{(x^{-m}-2^m)}$$
04

Evaluate the limit

Using L'Hopital's rule to evaluate the limit (since the limit is of the form 0/0), we differentiate the numerator and denominator with respect to x. Numerator derivative: \(-mx^{m-1}\) Denominator derivative: \(mx^{-m-1}\) So the limit becomes: $$\lim_{x \to 0.5} \frac{-mx^{m-1}}{mx^{-m-1}}$$
05

Simplify and solve for m

Simplify the expression as x approaches 0.5: $$\lim_{x \to 0.5} x^{2m} = 0.0625$$ Since we are dealing with limits, we can plug in x = 0.5: $$(0.5)^{2m} = 0.0625$$ Now, we solve for m: $$2m \log(0.5) = \log(0.0625)$$ $$2m = \frac{\log(0.0625)}{\log(0.5)}$$ $$m = \frac{1}{2} \cdot \frac{\log(0.0625)}{\log(0.5)}$$ The calculated value of m is 2. So, the correct option is (b) 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.

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

Let a function \(\mathrm{f}\) be defined by \(\mathrm{f}(\mathrm{x})=[(\mathrm{x}-|\mathrm{x}|) / \mathrm{x}], \mathrm{x} \neq 0\) and \(\mathrm{f}(0)=2\), then \(\mathrm{f}\) is: (a) Continuous no where (b) Continuous everywhere (c) Continuous for all \(\mathrm{x}\) except \(\mathrm{x}=1\) (d) Continuous for all \(x\) except \(x=0\)

\(\lim _{\mathrm{X} \rightarrow \pi}[\\{1+\operatorname{Cos}([2 \mathrm{~m}+1] \mathrm{x})\\} /\\{1+\operatorname{Cos}([2 \mathrm{n}-1] \mathrm{x})\\}]=?\) (Where \(\mathrm{m}, \mathrm{n} \in \mathrm{N}-\\{1\\})\) (a) \([(2 m+1) /(2 n-1)]^{4}\) (b) \([(2 m+1) /(2 n-1)]^{2}\) (c) \([(2 n-1) /(2 m+1)]^{2}\) (d) \([(2 n-1) /(2 m+1)]^{4}\)

\(\lim _{\mathrm{x} \rightarrow(\pi / 8)}(\sin 4 \mathrm{x})^{(\tan ) 2(4 \mathrm{x})}=\ldots \ldots .\) (a) \(\mathrm{e}^{(1 / 4)}\) (b) \(\mathrm{e}^{[(-1) / 2]}\) (c) \(\mathrm{e}^{[(-1) / 4]}\) (d) \(\mathrm{e}^{(1 / 2)}\)

If \(\mathrm{f}(\mathrm{x})=\mid \begin{array}{ll}\mathrm{m}+3 \mathrm{nx}, & \mathrm{x}>1 \\ 11, & \mathrm{x}=1 \\ 5 \mathrm{nx}-2 \mathrm{~m}, & \mathrm{x}<1\end{array}\) is continuous at \(\mathrm{x}=1\) then \(\mathrm{m}=\ldots .\) and \(\mathrm{n}=\ldots . .\) ? (a) \(\mathrm{m}=2, \mathrm{n}=-3\) (b) \(\mathrm{m}=-2, \mathrm{n}=3\) (c) \(\mathrm{m}=2, \mathrm{n}=3\) (d) \(m=3, n=3\)

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) \(\left(\ell^{2} / 16\right)\left(a^{2}-b^{2}\right)\)
See all solutions

Recommended explanations on English Textbooks

Language Analysis

Read Explanation

Textual Analysis

Read Explanation

Graphology

Read Explanation

English Language Study

Read Explanation

Text Comparison

Read Explanation

Listening and Speaking

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