Hey, everyone. We just learned how to find the limit of a rational function whether the denominator is 0 or not. So let's go ahead and work through these examples here. The first limit that I'm asked to find is the limit as x approaches 1 of x3-2x2+xx-1. Now, looking at what x is approaching here, 1, if I plug that into my denominator, I get 1-1, which gives me 0.
Now, because my denominator would be made 0 by that x value of 1, that means that I need to go ahead and factor this rational function. In order to find this limit, I need to fully factor my function as x approaches 1, and I'm going to factor that numerator. The first thing that I see is that each of these terms has an x in it, so I can factor that x out. That will give me x(x2-2x+1).
Since my denominator cannot be factored any further, that remains as x-1. I can continue to factor further because this polynomial here actually ends up being a perfect square. So I can factor that as x(x-1)2. This is still divided by x−1, and I am still trying to find the limit as x approaches 1. I see that I can cancel some stuff out here because I have x-1 on the bottom and I have x-1. I actually have two of them on the top. So that's no longer squared in my numerator. What I'm left with here is to find the limit as x approaches 1 of x(x-1), having canceled only one of those x-1 terms in the top with my x-1 on the bottom. This will end up giving me 1(1-1), which is 0.
Looking at our next example, we want to find the limit of the same function, but this time as x approaches 2. Now, if I were to plug 2 into my denominator here, that gives me 2−1, which is equal to 1, not 0. So I can actually just go ahead and plug 2 into my function here and directly substitute that in, not having to worry about factoring or canceling anything out because my denominator is not 0. Now doing this algebra here, 2x3-2⋅x2+22-1, gives me 2 over 1 or just 2 for the limit of this function as x approaches 2.
We now know how to find the limit of any rational function whether the denominator is made 0 or not. Now, if our denominator is 0, we know that we need to factor and then cancel our common factor out and then plug our value in. If our denominator is not 0, we can just go ahead and plug that value in from the beginning. Let me know if you have any questions. I'll see you in the next one.