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Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ G(x) = \dfrac{x^2 + 1}{2x^2 - x - 1} $

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02:21

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Harvey Mudd College

University of Nottingham

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

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.

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Explain, using Theorems 4,…

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$25-32$ Explain, using The…

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$21-28$ Explain, using The…

01:05

Okay we're given a function. And the the initial question is um why is it continuous at every point in its domain? And then it has to find the domain. Well first you've got to make sure you know what domain means. So domain is the set of all numbers X. That you can put into this function so that a real number comes back out again. So is there any way you could put a number in there? A real number? So that an imaginary number comes out something with? I. No. Okay so that's not a problem. The problem is what if you put a number in and you got this? So some number over 0? Well some number over zero is undefined. That's not a number. Okay. So if there are any numbers that caused the bottom to be zero then they are not in the domain of this function. And if you think about graphing, if it makes the bottom zero then there's a vertical sm tote there. Okay and that's a that's a line that the graph can't cross. So to find the domain of this function you have to find out what X. Is. Make the denominator zero and then leave those out. Okay. Like this. Yeah. So I bet I can factor the bottom. If I can't I have to use the quadratic formula on it. Um Two X x minus one plus one. That gives me two X squared minus two X plus x. Okay that'll work. Okay. two x plus one X minus one. You can either say can't equal zero here or equal zero here because you're going to use the information in a minute. So X equals want and X equals minus one half. So domain is all real numbers except X equals minus one half And x equals one. So as long as you don't put those two numbers in there then this function is fine. But if you put those one or the other of those in there then this function causes this undefined thing to happen and so that those two things are not in the domain. Okay. Here's another way to write it because I'm sure it doesn't look like that in your book and it's easy if you put it on a number line. So it's all the numbers starting at minus infinity Up to -1/2 but don't include it together with so I'm gonna put a union sign there All the numbers on this side of -1 half. Up to the numbers up to the # one, so minus one half to one Together with all the numbers on this side of one up to positive infinity. Okay. Got says leave out minus one half and one because the round bracket means don't include. Okay, so this and this say exactly the same thing. All right. Now, why is this function continuous at every point in its domain. Okay, um let's look at this one. two x plus one is a line, X -1 is a line. Those two are continuous. When you multiply continuous functions you get a continuous function. Yeah, X squared plus one is a parabola that's continuous. When you divide a continuous function by a continuous function, it's continuous. So this function is continuous everywhere there is defined. Okay, so the rules I used, I don't know what numbers they are in your book, but that product product of continuous functions is continuous and quotient of continuous functions is continuous and um the fact that each of these three are continuous because they are, I don't know if you have one that's Paulino meals are continuous. Okay, hope that help.

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