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Problem 20 Easy Difficulty

Explain why the function is discontinuous at the given number $ a $. Sketch the graph of the function.

$ f(x) = \left\{
\begin{array}{ll}
\dfrac{x^2 - x}{x^2 - 1} & \mbox{if $ x \neq 1 $} \hspace{40mm} a = 1\\
1 & \mbox{if $ x = 1 $}
\end{array} \right.$

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Video Transcript

uh When X is not equal to one F of X is defined to be X squared minus X Divided by Expert -1. Uh When X is equal to one F of X is simply defined to equal the number one, let's look at this top portion for the definition of F of X. Um Here's clearly why X is not allowed to equal one and actually extra not be allowed to equal negative one either. When X is one ah X squared one squared is one minus one is zero. So when X is one, this denominator would be zero and you cannot divide by zero. The same thing would happen. Uh Effects was equal to negative one. Effects was negative one. X squared negative one squared would be positive one and then minus one would be zero again. So this should say for x not equal to one or negative one but we won't get into that. Um This is saying, you know, for x not equal to one are when X is anything other than one. Here's the definition for ffx and when X does equal one uh Here is what Quebec's equals. Now we want to talk about why the function is discontinuous when x equals one. Well, I have this function graft. So let's go ahead and take a look at this. Great. Now uh the black curves are the graph of the X squared minus X over X squared minus one. Uh definition of the graph. And then this green point is when the function was defined to equal one. When XS one, one thing I want to note uh First is even though uh this black curve here looks continuous when X is one, there looks like there's a point when this black curve um There should be an open circle right here, but this most does not put those in. Uh So this black curve portion of the graph, the x squared minus x over x squared minus one. It's not defined when X equals one. When X equals one, there should be a little open circle on the black portion of uh this curve. And that's because uh this is not defined when X is one because one squared minus one will give you a zero in the denominator, you cannot divide by zero. So this is the graph of the entire function F of X. Uh They gave it to us in two different parts when x isn't equal to one when x is equal to one. Uh So what you see here, the black curves along with the green dot is the graph of the function. The only thing I want you to remember is that when X equals one, there really should be an open circle on this portion of the black curve. Now, why is this function F. Of X? Discontinuous when X equals one? Well, when X is approaching one from the negative side from the left side, Uh the curve is approaching the value or the functions approaching the value of 1/2 when X approaches one. So here's one on the X axis. When x approaches one from the right side or the positive side, as we approach X equals one from the positive side. The function is once again approaching the value of 1/2. But the function when x equals one has the value of one and this is why the function cannot be continuous. The limit of the function as X approaches one from the left side and the right side is one half. So the limit of the function as X approaches one is one half. But the value of the function when X is exactly at one is 1. The function cannot be continuous if the function at the point is not equal to the limit of the function as X approaches the point, let me go back to the white board here and show you what I mean. Okay, F is not continuous. We're going to say f is not continuous if it's not continuous at X equals one because F of one F evaluated when actually goes one is one. That comes from this portion of the function definition right here, but the limit of F of X As X approaches the value of one is one half. We got that from the graph. So the limit of the function as X approaches one is one half that happened from the left side and the right side of one, but the function when x equals one is the number one. So The limit of the function when x approaches one is not the same as the value of the function when X equals one, and that's why F cannot be continuous in order for it to be continuous. The limit of the function as X approaches to value has to be equal to the value of the function at that number. So As X approached one the limit of F of X as X approached one would have had to have been equal to the value of F at one, but it was not and that's why it's not continuous.