We just saw that for functions like polynomials and basic roots, our limit is always the same as our function value. We also saw that this was true for rational functions. The limit is the same as the function value as long as our denominator is not equal to zero. But from working with rational functions in the past, we know that this won't always happen. Sometimes our denominator will end up being zero. So how do we find the limit in these cases? Well, we're going to have to do a couple of extra steps, but I'm going to walk you through exactly what this process looks like. So let's go ahead and jump in.

Now with our rational functions where the denominator is not equal to zero, like this one here, finding the limit of x2+3x+2x+1 as x approaches 0. Since that 0 does not make our denominator zero, we can just plug it in everywhere and evaluate our function there and end up with our final answer. But what if we want to take the limit of this same function, x2+3x+2x+1, but this time as x approaches negative 1?

Well, the first thing we want to do here is just plug that value into our denominator. So if I plug that negative one into my denominator, I get negative one plus 1. Now negative one plus 1 is zero. And because my denominator is zero here, I can't just plug this negative one in and evaluate my function there. I have to do something else. So whenever we're faced with this, whenever our denominator is zero, the first thing that we want to do is factor the top and the bottom of our rational function. So let's go ahead and do that here. Let's go ahead and factor this numerator x²+3x+2. Now remember when factoring, we want to look for two numbers that are multiplying to 2 and to 3. So that ends up factoring to x+1 times x+1. And I have fully factored this rational function. So from here, what we want to do is we want to cancel out our common factor. We are always going to end up with a common factor and here we have x+1 in the top and x+1 in the bottom so I can go ahead and cancel those out. Now I'm just left to find the limit of x+2. So here I can proceed in evaluating my function as normal, just plugging this negative one in. So that ends up giving me negative 1 plus 2, giving me my final answer of 1 for the limit of this rational function as x approaches negative one.

So when you're faced with finding the limit of a rational function the first thing you want to do is check if that x value is going to make your denominator zero. If it's not going to make your denominator zero you can just evaluate your function plugging that x value in. But if it does make your denominator zero you want to start first by factoring the top and the bottom of your rational function. Then you can go ahead and cancel out the common factor that you end up with. And then finally, you can proceed in evaluating as normal. So with this in mind, let's work through a couple more examples here.

Now looking at this first example, we want to find the limit of this function x2+2x-15x-3 as x approaches 3. Now remember the first thing we want to do here is check if this value will make our denominator zero. If I plug this into my denominator, I get 3 minus 3, which definitely is zero, which means that I need to go ahead and start here by factoring. Now we're still finding the limit as x approaches 3, but having factored this numerator, remember that I'm looking for 2 values that multiply to negative 15 and add to positive 2. Now something that you may notice here is that because we are always going to end up with a common factor, I already know that one of my factors is just going to be this x-3. So that can make it sort of easier to factor and be kind of a shortcut for you here. Now I have this x-3 factor. And the other factor here is going to be x+5 for that numerator. Now in my denominator that's staying the same, it's still x-3. And my second step here is to cancel that common factor, which of course here is x-3. So that will all cancel out and now I'm just left to find the limit as x approaches 3 of x+5 because that's all I'm left with here. Now I can evaluate as normal just plugging in this x value of 3 to give me 3+5 for my final answer of 8 for the limit of this rational function as x approaches 3.

Now let's look at one final example here. Here, we want to find the limit of x+2x2-x-6 as x approaches negative 2. So again, the first thing we want to do here is just plug in that negative 2 to our denominator and check if it's going to make it zero. Now doing that here, I get negative 2 squared minus negative 2 minus 6, which ends up giving me 4 +2 minus 6 which is equal to zero. So from here we can't just plug that negative 2 in everywhere, we need to go ahead and factor. So my numerator here actually cannot be factored. It's just going to remain that x+2. So keeping my numerator the same there but factoring that denominator x²-x-6, I'm looking for two numbers that multiply to negative 6 and add to negative 1. But remember, I can kind of cheat a little bit here, take a shortcut, because I know that one of my factors is going to be x+2 because it will end up canceling out with that x+2 in the numerator. Now my other factor here is going to be x-3 in order to multiply back to what my original denominator was. Now from here, I have my common factor, that x+2, in the top and the bottom. That's going to end up canceling out here. And I am left to find the limit still as x approaches negative 2, but all I'm left with here is 1 over x-3. Now this negative 2 will not make this denominator zero, so I can go ahead and just plug this in as normal, giving me 1 over negative 2 minus 3, which ends up giving me a final answer of negative one over 5. So this is the limit of this rational function as x approaches negative 2.

So now that we know how to find the limit of any rational function, whether a denominator becomes zero or not, let's continue getting some practice. Thanks for watching. Let me know if you have any questions.