Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Show $\lim _{x \rightarrow a} f(x)=f(a)$ is equivalent to $\lim _{h \rightarrow 0} f(a+h)=f(a)$(Hint: let $x=a+h$ )

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Missouri State University

Campbell University

Baylor University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

01:54

$$\begin{aligned}&…

01:44

Prove that $\lim _{x \righ…

01:05

Find $$\lim _{h \rightarro…

01:12

01:22

Prove the limit statement.…

03:03

Evaluating a Limit from Ca…

01:56

Find $\lim _{h \rightarrow…

01:37

01:29

01:33

all right, This problem is another proof problem. So they've given us the answer. And the cool thing is, they're basically saying, you know, the limit of the difference question equals with limit of the slope of the Sikh in line as X approaches A. So the hint here was the let H equal X minus A. To show that this is indeed true. So we're gonna keep the limit as age approaches zero for just for just a moment. What we will change it and just a little bit. But I wanted to address what's happening in the actual function for just for to address that first. So if H is equal to X minus a, I should get a plus. And instead of h, I'm going to write X minus a. And then there's no age to substitute over here. And if H is equal to X minus A instead of writing h, I will write X minus a. All right, So what's happening in the first bit? Um, more going to continue to drop the right side down a plus X minus. A is just f of X, and then I have of a all over X minus. Hey, so I'm pretty closed. The only thing I need to do now is it. Toe Show that, you know, if H is approaching zero than X minus, A is going to be approaching zero. Right, And that could be kind of equal to zero or X. Could be a over X could approach A is kind of how you look at it. So now I can write that X approaches A. And I've indeed verified that the limit of the difference quotient could be rewritten as this limit of the slope of a Sikh in line as the two numbers approach each other all right and again, these both worked or both identical pretty much, and this is just one way of writing it against another.

View More Answers From This Book

Find Another Textbook

01:47

Use the appropriate rules to determine the derivative.$$w=32 v^{1 / 4}-\…

04:30

Find the point on the line $y=2 x+3$ that is equidistant from the points $(-…

02:14

Determine the equation of the tangent line at the indicated $x$ -value.$…

01:30

Let $f(x)=x^{-1 / 2}$ on the interval (0,1] . (a) Does $f$ satisfy the condi…

01:46

Find the instantaneous rate of change of $y$ with respect to $x$ at the give…

07:25

A Norman window is one in which the window is constructed by capping a recta…

02:40

A large cube of ice is melting uniformly at the rate of 6 cubic inches per s…

04:13

Use the first derivative to determine where the given function is increasing…

03:22

A spherical balloon is being deflated at a rate of 10 cubic feet per second.…

01:57

A 12 inch piece of wire is to be cut into two pieces. One piece is to be use…