Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Creating functions satisfying given limit conditions Find a function $f$ satisfying $\lim _{x \rightarrow 1} \frac{f(x)}{x-1}=2$.

Calculus 1 / AB

Chapter 2

Limits

Section 3

Techniques for Computing Limits

Campbell University

Harvey Mudd College

Boston College

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

01:17

Find a function $f$ satisf…

06:47

Creating functions satisfy…

10:53

Compute $f^{\prime}(x)$ us…

0:00

Find a formula for a funct…

01:07

Assume $\lim _{x \rightarr…

03:53

00:20

01:00

Find two functions $f(x)$ …

07:51

Find the limit $\lim _{x \…

03:10

Find the limit of $f$ as $…

here. We're trying to find a function that satisfies this equation. So we're looking at the limit as X approaches one of F of X, divided by X minus one, and that has to be equal to two. So our job is to find an f of X such that this limit expression is true. Now, there are a couple different ways to do this, but my recommendation to use that the easiest way to do this is to find an f of X where we have an X minus one included. And then from there, the rest of sort of sorts itself out because we can cancel the X minus ones. So I'm going to choose F of X to be X squared minus one, which we know is equivalent to X plus one x minus one. They might be wondering, Why did you choose F of X as X squared, plus one or minus one? Excuse me. And the reason for that is because then the X minus ones will cancel and we're left with X plus one. And when we plug in our value, we get one plus one, which equals two. We'll go ahead and verify this. We get the limit as Ex Purchase one of F of X, which will choose to be X plus one x minus one divided by X minus one equals two. And so we'll go ahead and cancel out our X minus one. So we're left with the limit as exit purchase one of X plus one, and then we can go ahead and plug in our X. And then we know that one plus one is two and to does equal to So are F of X that we chose. X squared minus one does work.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

Find a function $f$ satisfying $\lim _{x \rightarrow 1}\left(\frac{f(x)}{x-1…

Creating functions satisfying given limit conditions Find functions $f$ and …

Compute $f^{\prime}(x)$ using the limit definition.$$f(x)=x^{-1 / 2}…

Find a formula for a function $ f $ that satisfies the following conditions:…

Assume $\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,$ and…

Compute $f^{\prime}(x)$ using the limit definition.$$f(x)=1-x^{-1}

Find two functions $f(x)$ and $g(x)$ with the given properties.$$\lim _{…

Find the limit $\lim _{x \rightarrow 1} f(x)$ and determine if the following…

Find the limit of $f$ as $(x, y) \rightarrow(0,0)$ or show that the limit do…

02:00

Determine the following indefinite integrals. Check your work by differentia…

03:23

Estimating roots The values of various roots can be approximated using Newto…

04:27

Definite integrals from graphs The figure shows the areas of regions bounded…

02:16

00:18

Use the guidelines of this section to make a complete graph of $f$.$$f(x…

02:09

Assuming $f$ is differentiable for all $x,$ simplify $\lim _{x \rightarrow 5…

10:39

Properties of integrals Consider two functions $f$ and $g$ on [1,6] such tha…

00:42

Explain how to use definite integrals to find the net change in a quantity, …

03:05

Using properties of integrals Use the value of the first integral I to evalu…

02:07

Approximate the change in the lateral surface area (excluding the area of th…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.