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

Use the results of the previous exercise to compute each of the following limits: (a) $\lim _{x \rightarrow 2} \frac{x^{3}-8}{x-2}$ (b) $\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}$.

(a) 12(b) 4

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Missouri State University

Baylor University

University of Nottingham

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.

06:27

Evaluate each of the follo…

01:37

Calculate each limit in Ex…

01:05

Evaluate each limit and ju…

00:54

00:19

In the following exercises…

00:40

For the following exercise…

02:42

Calculate each of the limi…

02:52

01:46

Determining limits analyti…

04:47

all right, So this problems ask you to do low P tells rule, Although you might not know it's called, Lucy tells rule eso part. They were doing limits extra approaches to of X cubed minus eight over X minus to And in order to do this problem, what you need to do is make sure that when you do direct substitution and plug into into the numerator, you get a value that's eight minus eight, which is zero. So you have to make sure you get zero there. Same thing with the denominator is you have to make sure that you get zero here. There's other forms of that. We can use the hotel's rule. Um, you know. But for the sake of this lesson, all you need to do is establish that you have 0/0, and then you can take the derivative of the top, which is three x squared over the derivative of the bottom, which is one which I hope doesn't confuse you because it's not the crucial role notch. Oh, shit hole. Very confusing. So we've been doing the closure rule in this lesson, and then they throw. Loki tells rule that you which is not, uh so now we can plug into and for X. When we do that, we get two squared US four and three times four is 12. So that's your first answer to part A. It is part A, by the way, on in part B, we're looking at X to the fourth minus one over X minus one or during the limit as X approaches. One is in part B really the same thing, though you want to just double check that when you plug in one to the fourth power minus one, you get zero on top and same thing with the bottom plugging in one minus one. You do get zero on bottom so you can apply low Patel's rule, which is then taking the drift of of the top, which is four x cubed. And then the directive of the bottom, which is just one and notice it is not a quotient rule. So as you work through that plug in one well four times one Q one, Cuba's one times four. We'll give you four. So part A is 12 in Part B is four, and there you have it

View More Answers From This Book

Find Another Textbook

Numerade Educator

01:35

Write the given expression in exponential format. $$\log _{2 / 3} 16 / 8…

04:36

Find $d y / d x$ using any method.$$2 / x-3 / y=x^{2} y^{2}$$

03:27

$1 / t+1 / s=1 .$ Find $(a) d s / d r ;$ (b) $d t / d s,$ (c) Show that $(d …

01:29

Sketch the graph of the function defined by the given equation.$$y=f(x)=…

02:01

Suppose postal requirements are that the maximum of the length plus girth (c…

01:57

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

01:03

Assuming each limit exists, compute them. (a) $\lim _{h \rightarrow 0} \frac…

02:56

Sketch the graph of the function defined in the given exercise. Use all the …

03:54

Find the equation of the tangent line to the curve at the given $x$ -value.<…

06:45

The Amalgamated Flashlight Company shows a profit of $4,500$ on a production…