 Welcome back to our lecture series math 1210 calculus one for students at Southern Utah University as usual I'll be a professor today. Dr. Andrew Missildine in this video Starting lecture 10. I want to talk about using limit laws to help us calculate Limits and that's because previously in this lecture series You've seen actually two ways to calculate limits the first one of these the precise definition of the limit using the Epsilon's and Delta's we can be very precise in our calculations, but That approach is very cumbersome especially for New folks to the realm of calculus. So although we did introduce this idea of error and tolerance beforehand We're really not going to pursue that precise definition much more in this lecture series Instead we've relied more on this intuitive notion where the limit is calculating What we expect the function to do not what it actually does and we saw in the previous lecture number nine Using graphical representations of functions exactly what that means to intuitively understand what this limit is but Graphical approaches although very nice and help strengthen our intuition. They do have limitations We have to have an accurate graph in order to be able to compute the the limits in that case And we also saw situations like the topologist sine wave where even the graphical approach is insufficient and could be confusing And so we want to in this lecture rely on an algebraic approach to computing limits Now all of the algebraic laws we're going to see in this video can be proven precisely for functions and This would come from a theoretical approach proofs relying on that Epsilon Delta definition of the limits So in this in this video, we aren't going to provide the proofs of these limit laws And just rely sort of on the intuition to fortify These statements for us. So these are some rules of limits that we should be aware of so in all of these limit laws follow the Following notation will be consistent here So let the number little a capital in capital B be real numbers You will notice that I'm using a little a and a capital a this is quite common in mathematics Where the variables are actually case sensitive if you're talking about the function little f That's not the same thing as capital F, which could be something very different So when you write your mathematical notation do make sure you are case sensitive like maybe your password Online or something like that. So little a capital in capital B are going to be real numbers little f and little g will be functions and Let the following statements be true the limit as x approaches little a of f of x will be capital a and the limit as X approaches little a of g of x will be B So you'll notice that in both situations these limits are as x approaches little a In order for these limit laws to be true. We have to be approaching the same value X x equals a right there Alright, so what are their first couple limit laws? The first limit law that we're gonna have tells us that if we take some value k which this is a constant So it won't vary like f of x does So if f of k is a constant then it turns out the limit as x approaches a of k is going to equal k in other words If you have a constant function which would just be a flat line and you're interested in what happens at x equals a Well, if you approach it from the right from the left, excuse me Everything's going to be k and as you approach from the right everything's going to be k So the limit's going to be k for a constant function nothing changes So you wouldn't expect it to suddenly change the expectation is constants would be k on the other hand If you take the limit as x approaches a of k times f of x So if you take a scalar multiple of a function Geometrically it has the effect of stretching the graph by factor of k if you were to vertically stretch a graph That means you're also going to stretch the limit So the limit of k times f of x is k times the limit of f of x that is k times a where a was the limit of the function so Times in a function by k will also times the limit and that does make sense as well If you have a function where your limit here is given as a and then you vertically stretch that function Then the y-coordinate will also be stretched by that factor of k Limit law B if we take the limit of f of x plus or minus g of x This will equal the limit of f plus or minus the limit of g which turns out to be a plus or minus B So the idea with this limit law is that that you if you take the limit of the sum of two functions This is a sum of limits or if you take the difference of two functions The limit will be the the limit or the difference of their limits So that is to say you can take individual pieces separately and we'll see this in just a moment if you're taking like the limit of Say like x squared plus 3x you can break this up into the limit of x squared plus the limit of 3x you can take the two portions Individually for which for the second part by limit law number one You can pull out the coefficient three and you get the limit of x So assuming we can compute the limit of x squared and x we can do this polynomial Combination and so that these properties right here are starting to show you how we want to algebraically approach these things One thing I do want to point out right here is if you look at properties a and b like we saw with the polynomial Property a tells us how we can take the limit of a scalar multiple We're scalar here There's a fancy word for a constant multiple and then property B tells us what we can do with with sums and differences of Limits when you put these properties together. This is what's known as the linearity property This is something we're gonna see over and over and over again in calculus limits are linear derivatives Which we haven't defined yet in this lecture series are linear Anti-derivatives sums that is sigma notation integrals both definite integrals and indefinite integrals Infinite series these are all linear operators. They satisfy the linearity property and it's for this reasons why when one gets exposed to calculus It becomes ultimately necessary for them to study linear algebra which is the study of linear things and I won't say much more about that in this lecture right here Let's continue with our rules of limits here Just like when it comes to addition subtraction take the limit of a product the limit of f of x times g of x This will become the limit of f times the limit of g and that becomes a times B And I should mention with these properties where you take the limit of some combination and break it up into individual limits This limit all this this limit property only exist if these limits exist by themselves It is possible that the limit of a sum could exist But the two summands don't exist the limits don't exist by themselves But of course under the assumptions we have right now the two summands or the two components do exist by themselves These limits and so therefore we can decompose these more complicated things via operations property D is the is the similar statement about Quotients here if you take the limit of f divided by g then this will be the limit of f divided by the limit of g Specifically a divided by B. There is one exception there if the limit of g is itself Zero like if B is zero then you'd get something divided by zero that doesn't guarantee that the limit exist This property can't it turns out the limit could exist it might not division by zero is a little bit But trickier is something we'll talk about at the end of this video a little bit more and then also in future videos as well So even though this list might seem imposing the properties listed here are actually quite natural and the viewer will grow To use them rather quickly right we will also mention that although we can prove the above properties about limits I you know I mentioned that earlier. We're not going to do so because again that kind of gets us into what's called real analysis the the proofs of Calculus things and that's that something that happened in a different lecture series But let's apply these to some examples Let's suppose that the limit of f of x as x approaches 2 is 3 and the limit of g of x as x approaches 2 is equal to 4 Notice that both limits are approaching 2 Even if you swap this up to be like 1.9, we wouldn't be able to combine these limit loss together They need to be approaching the same value So the limit of f is 3 and g is 4 as x approaches 2 So when you see something like the limit as x approaches 2 of f of x plus g of x This would tell us something like oh, this is the limit of f of x plus the limit of 5 times g of x Like so as x approaches 2 in both situations with the limit of f we know that's equal to 3 and We get this we got we got this Decomposition here by law number 2 we saw on the previous slide and then Since you have this constant multiple of 5 the first law says you bring out the 5 and take the limit of G of x as x approaches 2 Which we see from assumptions. That's 4 how that's equal to 4. We don't know it just we just suppose we know that we get 3 plus 5 times 4 we're going to get 3 plus 20 and so the limit would be 23 Based upon the limit loss that we saw previously If we take the limit as x approaches 2 of 2 times f of x times g of x Well, you can bring out that too so we get the limit of f of x times g of x as x approaches 2 And then by law number 3 we saw in the previous slide You're going to get 2 times the limit of f times the limit of g of x Again as x approaches 2 in both situations for which then by assumption We see that the limit of f was a 3 and the limit of g was a 4 So we end up with 2 times 3 which is 6 6 times 4 is 24 and that would be the limit in that situation All right, so what do we do on this next one here? So we have a limit of f of x squared time are divided by the natural log of g of x Well, the fourth law that we saw in the previous slide would tell us that this should be the limit as x approaches 2 of f of x squared over the limit As x approaches 2 of the natural log of g of x So that was using law d we saw in the previous slide, but what do we do past this right? So we know that the limit of F is equal to 3 But what about the limit of f squared and we know that the limit of g is going to be 4 but without the natural log of g We don't it turns out this example illustrates that we don't have enough limit properties And as it is in order to to tackle all of the type of problems one might see all the types of functions One might see from pre-calculus. So let's add to our limit laws and So picking up where we left off Property e tells us that the limit as x approaches a and so this a is any any real number, right? Remember the same assumptions we had when we started this here So as x approaches a x to the n is going to approach a to the n for all positive integers in right here So if n is any monomial you'll see that The limit as x approaches a of x to the n is just going to be a to the n You'll notice here that this is just function evaluation to calculate the limit of a monomial. You just have to Evaluate the monomial at this number inside of its domain and so I want to point out that by this property e and The limit laws we saw previously those linearity properties a and b when you combine those together This actually implies that we can compute the limit of any polynomial function just by evaluating the polynomial And we'll actually see in a forthcoming example an example of such a thing right there All right, so for any real number k If you'd have the limit of f of x to the k This is kind of like we had it before right take the limit as x approaches a of f of x to the k This is equal to the limit of f of x to the k Right so you can just take the limit of f which is a and then you take a to the k so you have some type of Some type of power expression where the exponent is fixed and the base is a variable in that situation You can just eval you can just take the limit of the base which that actually has a lot to do with what we were doing a moment ago That's exactly the setting we were in right here this limit law when applied tells us that Because we taking the limit of a function to an exponent This will just be the limit as x approaches to of f of x quantity squared For which we saw that the limit of f of x was equal to three so this will become a three squared a k a nine All right, but what do you do about the denominator? We don't have enough information. What how's the natural log effect the limit here? Let's look a little bit more in our list So in addition to property f which guarantees that power functions You can just take the limit of the base for any real number B, which is positive Take the limit as x approaches a of B to the f of x notice the difference here in this situation The base of the exponential expression is now a constant That's why we require it to be positive because if we live negative bases then that Potentially could give us some imaginary numbers. We don't want that the base being zero is also a little bit more complicated We'll deal with that in a future video Oftentimes it's gonna be zero, but there are sometimes where it might not be zero again That's a topic for another another lecture But in this situation if you have a just have a simple base just a positive base where the exponent is some type of variable It could just be x it could be x squared or something more complicated in that situation when your base is constant You can just take the limit of the exponent and so the limit would be B to the a like so that doesn't help us out with our logarithmic expression But then when we get to property h, that's exactly it if we take a positive base If we take a positive base B, that's not equal to 1 I noticed that it says B's positive twice there. I'm not sure why that's necessary Probably just to type out. Sorry. Anyways, if we just take an acceptable base for a logarithm If you take the limit as x approaches a of log B of f of x This will become log B of the limit as x approaches a of f of x that is you'll be the logs base B of a So if you have a logarithm inside of a limit, you can actually take the log outside and just evaluate the inner function Take the limit in that situation Same thing is also true for sine and cosine if you take the limit of sine of f of x This becomes the sine of the limit of f of x and if you take the limit of cosine of f of x This will become cosine of the limit of f of x now If we can do if we can take that and combine it with trig identities like the quotient identities and the reciprocal like the identities That is sine and cosine generate all the other trig functions like tangent sine over cosine C Kansas one over cosine Since we can take limits of quotients and we know the limit of sine and cosine We can then use this to take limits of other trigometric functions as well And so these properties right here that we see that basically you can just evaluate All right, sorry, you just take push the limit inside of the function This has this is a property which we refer to as continuity Which is a topic we will talk about in the next lecture in much more detail But the typical functions we know about Exponentials logs trigonometric functions algebraic functions. These are continuous functions on their domains Like I said, we'll talk some more about that in the future So returning to our question at hand here by the previous property property H We see that the limit of the natural log of g of x is x approaches to this will then become The natural log because the natural log is just base e. It's the log base e So this would be the limit as x approaches to of g of x like so for which then g of x It's limit remember turned out to be four So we're going to the natural log of four, which if you want to you can write that as two natural log two Although that simplification is not really necessary in the situation, but the final limit would be nine over the natural log of four Let's look at a few more examples and then Finish this video right here. I mentioned previously that the laws of limits that we've seen already Give us the tools necessary to take the limit of any of any polynomial function So take for example 2x 2x squared minus 3x plus 4 take the limit of that function as x approaches 5 So by previous properties we saw that if you have the limit of a bunch of sums and differences You can actually break it up into individual sums and differences That is the limit of 2x squared minus 3x plus 4 breaks up to be the limit as x approaches 5 of 2x squared minus the limit as x approaches 5 of 3x Plus the limit of 4 as x approaches 5 Okay, so we saw that from property 2 in our list then you have some coefficients So we have like the limit of 2x we saw by property One property a in our list that these constant multiples can come out This comes two times the limit of x squared as x approaches 5 Then we also get minus 3 times the limit of x as x approaches 5 and property a also told us what to do with the constant function if you take the limit of a constant function Y equals 4 the limit will be that constant value So the limit of 4 as x approaches 5 will just be 4 because 4 doesn't change as x gets closer and closer to 5 There's no variability there. So we would expect it to stay 4 Then what do you do with these monomials? Well, we saw in the previous slide by law number e that if you have a monomial you can just evaluate it So the limit of x squared as x approaches 5 will just be 5 squared So you get 2 times 5 squared and then for the next one you're going to minus 3 times the limit of x Well, x is just the monomial x to the first and where for we're just going to take 5 to the first a k a 5 You get plus 4 right here. And so simplifying this calculation. You're going to get 2 times 25 Minus 3 times 5 which is 15 plus 4 you get 2 times 5 which is going to be 50 Negative 15 plus 4 is going to be negative 11 and so then we take 50 minus 11 and we end up with 39 So that's going to be the limit here. I want you to be aware because of this statement right here 39 is just the evaluation of this polynomial at 5 So if we call this polynomial say f of x right here Then 39 is just f of 5 when it comes to a polynomial we can evaluate it because polynomials are continuous again Something will define in the next and in a future video What if we want to take the limit as x approaches negative 2 of x cubed plus 2x squared minus 1 over 5 minus 3x? Well, what we've learned previously is that when you have this quotient We're going to take the limit on the top and the limit on the bottom. That was law number d so take the limit as x approaches negative 2 of x cubed Plus 2x squared minus 1 and then take the limit of the bottom We're going to take the limit as x approaches negative 2 of 5 minus 3x You'll notice that the numerator is a polynomial. So its limit will just be evaluation It'll just be negative 2 cubed plus 2 times negative 2 Squared minus 1 the bottom is also a polynomial 5 minus 3x So its limit will also just be evaluation at negative 2 5 minus 3 Times negative 2 like so and you'll notice that hey, this is a rational function This is a quotient of two polynomial functions by the same reason So by since polynomials limits can be computed by evaluation It turns out that rational functions can also with the potential exception of Those places that make the denominator go to zero, right? Because law d didn't doesn't work if the limit of the bottom is zero Which the good news here if we evaluate the denominator first you're going to see that you get five Plus six negative three times negative two is plus six that's going to equal a positive eleven And so the limits going to exist here. Let's continue on we get negative two cube Which is a negative eight negative two squared is a positive four times that by two That's going to give you a positive eight and you get a negative one We see that the eights cancel out you're left with a negative one And so that will be the limit here negative one eleventh