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Problem 23 Easy Difficulty

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} $

Answer

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

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Video Transcript

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.