All right. Welcome. Let's do some examples in this lecture. This is going to be examples that mimic the homework that mimic the test. The way to get better at working with limits is to just do some problems. Let's do them. Let's start off with the limit as x goes to 5 of x squared minus 6x plus 5 all over x minus 5. Here we go. This is going to model our approach. This is the goal of calculus here to become better problem-solvers. The first thing I want to do when I get a problem is not to panic but let's think about what we can do. What I think, the first thing we should try, is can we plug in? Can I just take five and plug it into this function? It's a rational function, I'm allowed to do so if I can, but you'll quickly see that if you plug in five into the denominator, you get zero downstairs, so you cannot. All right, but at least we tried. Then you look at this thing and you remember you taking years of math courses, you have some knowledge here what to do. That numerator should look like there's something you can do. You've seen this before. What can you do with that numerator? Well, you have a quadratic, you can factor it. You can factor, and when you do that, you get x minus 5x minus 1. Now maybe you can see the benefit of factoring this polynomial, the x minus 5 cancel. How nice is that? Then you just get left with the limit as x goes to 5 of x minus 1. Now I can plug in, and of course, that's just four. I'm going to put this over here, let's keep this over here, our ideas, our techniques to problem-solve. Let's call this factoring. Now, of course, not everything is going to factor, but that's a good one. Plug in didn't work. We try and we try again until something does. Let's do another one. How about the limit as h goes to 0, why not? Don't get used to just x, 4 plus h squared minus 16 over h. Well, let's plug in. Can we plug in? If I plug in the numerator, that's all fine, denominator, that's a problem. If you plug in, you get 0 over 0. Remember, don't make me mad, don't say, "Aha." This is one. No. This is nothing. This is undefined. You'll make me angry if you say, "Isn't that just one?" No. Okay, can I factor this thing? No. But there's something else that it's screams to do. When you have 4 plus h squared, what should you try here? How about foiling? So limit as h goes to 0, first outside, inside last, 16 plus 8h plus h squared minus 16 over h. I'm going to put foil here on our list of things to try. Is this good? Something is canceling, which is promising because I have the 16s are canceling and then if you write this, you get the limit as h goes to 0, there's an h actually left in both terms so I'm going to factor that out as well. See there's an 8h and then h squared, factor that h out, cancel the h's and you get the limit as h goes to 0 of 8 plus h. I can do that one. You just plug in, you get eight plus 0. Now notice I couldn't plug in before. This is the first time I actually can plug in. You're going to plug in when you can, as soon as you can, and you just couldn't do it before because that h on the nominator and you get good old eight. Here's an example when that 0 over 0 turns out to actually equal 8. This is why it'd be mistake to just put it like, "Is this not always one?" No. Sometimes it actually will be one, but there's no reason, or there's just an undefined expression. That's another example and just something else to try to do. Let's do another one. By the way, I want you to notice something before I move on. See how here I'm writing the limit every single time? You have to write the limit every single time. Do I have to write the limit every single time? Yes. Write the limit every single time. A lot of times if I give one of these, it just disappears, and that's a little bit of a problem. Now it disappears because you're focusing on the algebra and you're probably not used to writing it, but you've missed the point. You've basically taken the piece of calculus that we're studying limits and just thrown in the garbage. You say, "Oh, this is just foiling." No, we're foiling to evaluate a limit. You have to write the limit every single time so that you remind yourself take a limit because only when you plug in, only when you evaluate the limit, does the actual limit go away. It tells me you have not evaluated this limit yet. So you have to write the limit every time, fight your urge to be lazy here, and write it and show all your work. Because technically it is incorrect if it just disappears too early. Here's another example. Let's do 1 over 4 plus 1 over x over 4 plus x. Can I plug in? Now the denominator here, 4 plus 0 is fine. Four plus 0 is 4. That's kosher, that's allowed. What about the numerator though? There's a problem, 1 over 0 in the numerator prevents me from just plugging in. Well, I tried. Can I factor this thing? Is that a quadratic or something that I can factor? No. Can I foil this is something squared? No. What should I do? Well, the numerator screens basically clean me up. Let's put this on here, I don't know how to call this one, like algebra cleanup. Just to add or subtract fractions, just do what they did not do. Let's try that. When you add these together, you get 4 plus x over 4x, so 4 plus x over 4x looks good, that's upstairs, and then 4 plus x, there it is. If you forgot how to add or subtract fractions, I use the Bowtie method. You go look that up if you want or you can do common denominator, whatever works for you. But pause the video and go look that up if you forgot or check that for me. Now I have what's called a complex fraction. I have a fraction divided by fraction. You don't see the bottom as a fraction and think of it as over 1. How do you deal with that? You have 4 over x, 4 plus x or 4 over x. Keep, change and flip. Keep the numerator, change division to multiplication, and then flip the denominator. All of a sudden you going to start to see that like, this is probably the right way to go because this 4 plus x is canceled. Notice I'm running the limit every time and notice I can't plug in just yet because I have 1 over 4x. Let's clean this up just a little bit, so I have 1 over 4x. Now remember the Limit laws say, you don't have to do this, I'm going to write this out. You can pull out constants. I just want to write this so it's something that we've seen. The one-fourth is there, you can keep it there, you can move it out. But at the end of the day, what this is asking is, what is the limit as x goes to 0 of 1 over x? This is why you have to know your graphs. I'm going to draw this graph. Where do I have room? Right here. One over x looks like that. High to low, low to high asymptote to 0. The limit as x approaches 0, as I approach 0 from the right, I go to positive infinity. As I approach 0 from the left, I go to negative infinity. The two one-sided limits differ, and so this limit does not exist. That's the answer for this one. One-fourth times x doesn't exist, does not exist. This limit doesn't exist. Right away get used to limits not existing, that will happen and that's a perfectly fine answer, just as 4 and 8 and all the other ones were. There's a couple good ones there. Let's try another one. Can keep your handy dandy little list of things to try like nearby. Let's do the limit. Yeah, limit. As x goes to 0 of 4 minus root x over 16x minus x squared. Remember our things to try. One, I want to evaluate a limit, can I plug in? Is that an option here? This is your thought process or practicing how to think, how to problem-solve, how to approach these problems. No, you can't plug in 0 because the bottom is going to give you 0 pretty quickly. I tried. Second one, can I factor anything out? I don't know. Maybe. There's an x in both of them. There's no bad ideas in brainstorm, so let's try, 4 minus root x, x 16 minus x squared. That's pretty good. Well, this one is tricky. Can you factor anything else? Stare at this for a second. The denominator is a difference of squares, 16 minus x squared, there should be an x here. That's just x, 16 minus x. That actually factors, that may not be immediate at first, but it factors, when you have a difference of squares, its factors is 4 minus root x, 4 plus root x. That's true for a difference of squares. That is tricky, not super-obvious, so look out for that, and now you can see why that's super good, because that cancels. That cancels, and what are you left with? You're left with the limit as x goes to 0 of 1 over x, 4 plus root x. Now what do you do? Well, this is good. The 4 plus root x, this becomes root 0, this goes to 0 and you basically get one-fourth. This piece goes to one-fourth, but you're still stuck with is one over x again. This denominator, as x gets small, the denominator gets really big, it's not going to go to a certain number. You have this one of x again. This is for all the same reasons. This will not exist. This graph does not exist. You can also graph it if you want to see it, but you can also see it algebraically. There's nothing left to cancel with the numerator, if you can't plug in 0, you're done at this point, does not exist. That's a tough one. Let's do another one. They are getting a little harder of course as we go on. Notice I wrote the limit every time. Limit t goes to infinity, square root of t squared plus 9 minus 3 over t squared. Let's go through our list of things to try. Plug-in, can't do it, bottom 0. Factor, no, nothing is going to factor here. What else was on our list from before? Foil. Can't foil. Algebraic cleanup. Like combined some fractions or something like that, is not an option? Not for this one. We need something else for this one. This one I'm going to tell you the trick for this one because it is not super obvious what to do. This is one of my favorite ones because it's like watching a magic trick every time. What color should we pick? Let's pick this color. The trick here, to do this is called the conjugate trick. It is used a lot when you have square roots. Here's what I want to do. I'm going to multiply the expression by the square root of t squared plus 9 plus 3. I can't just multiply that by itself. I'm going to multiply top and bottom. Whatever we do to one, you have to do the other. Whenever you have a fraction with the numerator and denominator equal, this is a fancy way to write the number 1. I'm like multiplying by 1. Hopefully you agree if I multiply by 1, I haven't changed the problem. I'm allowed to multiply by 1. It seems crazy to do this, but bear with me, it's going to work out. I'm going to multiply by the conjugate. What's the difference? Just notice the conjugate I change the sign outside of the square root. Why would you do this? Well, watch what happens. I'm going to write the limit again. Always write the limit. Do I have to read the limit? Always write the limit. The top becomes a FOIL. First Outside, Inside Last. But here's the beauty of the conjugate. When you FOIL something and it's conjugate, the two inside terms, you get 3 root t squared plus 9 minus 3 root t squared plus 9. They cancel. That will always happen when you FOIL. The first term becomes square root of t square plus 9 times square t square plus 9. That's t squared plus 9. Outside, inside cancel. Again, if you don't see that immediately work that out, you'll see that those two cancel. Outside, inside cancel, and then the last becomes minus 9. The denominator, this is your cost of doing this trick. Nothing really fancy happens here. Leave it separate. Don't distribute, t squared plus 9 plus 3. Just two pieces multiplied together. Remember fraction times a fraction, multiply across the top, multiply across the bottom. You say, I'm not sure why this is any better, we'll stuff starts to cancel. The 9s cancel immediately and you're left with t squared upstairs. Then there's a t squared downstairs. The t squares cancel as well. A lot of stuff cancels. That's why this trick is so powerful. It really cancel some stuff. The beauty of it is, remember the problem that was preventing me from plugging in the beginning, this t squared and the denominator, it just went away. It absolutely just went away. Let me write this over. This is the limit as t goes here with what's left. In the numerator, everything canceled. You got to appreciate that for a minute. The denominator, you are left with t squared plus 9 plus 3. T square plus 9 is all under the square root and plus 3. Notice I wrote the limit again because now I'm going to evaluate this. When this t squared goes to 0, you're left with, and I plugged in, so the limit goes away. One over the square root of 9 plus 3, which of course is one-sixth. By the way, in a prior lecture, this was the one that we made the table for that we guessed, but it was looking to go to one-sixth. Now, we actually showed that it was one-sixth. This conjugate method is very powerful and use with square roots, so keep it in mind. Here are five techniques, plug-in, factor FOIL, algebraic, cleanup, and conjugate, that will be very handy when you start doing problems from this section and try to evaluate them. There are more to add to this list and we'll talk about that in future videos. Go over these and make sure you understand at least these techniques before moving on.