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How would you "remove the discontinuity" of $ f $? In other words, how would you define $ f(2) $ in order to make $ f $ continuous at 2?

$ f(x) = \dfrac {x^2 - x - 2}{x - 2} $

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$f(x)=\frac{x^{2}-x-2}{x-2}=\frac{(x-2)(x+1)}{x-2}=x+1$ for $x \neq 2 .$ since $\lim _{x \rightarrow 2} f(x)=2+1=3,$ define $f(2)=3 .$ Then $f$ is continuous at 2.

02:15

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

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University of Nottingham

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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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we look at the function f of x equals x squared minus x minus two over x minus two. You can see that this function is not continuous. We're not even defined when x equals two because effects is to uh in the denominator you have two minus two which is zero and you can divide by zero. Uh So the function is not defined when x equals two and obviously obviously not continuous one, X equals two. How can we remove this? Dis continuity? Uh Well we can do it by factoring if we factor x squared minus x minus two? Uh We get x minus two Times X-plus one X plus one. Uh So double check. Using foil, X squared minus two, X plus one, X is minus one, X negative two times one is negative two. So x squared minus x minus two Factors into X -2 times x plus one. This is being divided by X -2. And so now you see um this is how we remove the dis continuity. Uh we since you have X -2 being multiplied uh in the numerator and that's also what we are dividing by denominator. We can cancel out the X -2s and we are left with X plus one. So that is how we remove the dis continuity. But now remember that ffx was not defined when X equals two. So now we have to define our function F when X is too and we want it to be so that the entire function will be continuous. Well F of X is equal to X plus one. Our original function F of X would equal X plus one. Uh We uh cancel, factored out and cancelled out the x minus twos. F of X is equal to X plus one. Except when X equals two. So this is what F of X equals except when X equals two. Now, ffx Okay, X plus one. What does ffx? What is the limit of F of X as X approaches to? What does this function approach? What is this value going to get near as X approaches to? Well, as X approaches to X plus one obviously approaches to plus one which is tree. And so to limit, let's squeeze it in here. To limit of our function F of X as X approaches to. Okay, well, the limit of F of X as X approaches to is two Plus 1, three. So, if we want to define F when excess too, so that the function is continuous, F of two has to be the limit of F of X as X approaches to. That's what we mean by uh continuity. The limit of the function as X approaches, the number two Has to equal the value of the function at two. So, since the limited function is three, we are going to define F of 22 B, three.

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