Hi, I'm Mark Barsamian with the math department at Ohio University in Athens, Ohio. This is the second in a collection of videos for a one-semester course in calculus. Here at Ohio University, it's our course Math 1350, Survey of Calculus.
In the first video, which was called Limits Video A, I introduced the concept of limits and then did examples of what I called limits from a graphical perspective, that is, the functions involved were described by graphs. not by formulas. In this second video, Limits Video B, I'll be again discussing limits, but this time from what I would call an analytic perspective, that is the functions involved will be described by formulas, not by graphs. Our topics will be first an example where we estimate the value of a limit, and then I'll introduce some tools that are theorems presenting properties of limits, and then we'll resume examples discussing examples of computing limits analytically.
Remember that means the function involved is given by a formula not by a graph. Now remember that to view or download the PDF of these notes you can follow the link provided in the description portion of the webpage for the YouTube video. We'll need to use some stuff from our previous video. We discussed the definition of limit.
It was this symbol spoken the limit as X approaches c of f of x is L and in that symbol x is a variable, c is a number, f of x is a function, and L is a number. What this symbol means is the following. As x gets closer and closer to c but not equal to c, the value of f of x gets closer and closer to L. And what that means in terms of the graph of the function is that the graph of f f appears to be heading for the location with x y coordinates c comma l.
So whatever the number c is that's sitting here and the number l that's sitting here those are the x y coordinates of a location on the graph that the function appears to be heading for. Now it's important to remember the distinction between f of c and the limit as x approaches c of f of x. The symbol f of c denotes the y value at a particular x value x equals c whereas this symbol the limit as x approaches c of f of X tells us about the trend in the y values as x gets closer and closer to c as I mentioned earlier in this video we'll take an analytical approach to limits that is the function f is given by a formula not a graph in our first example we'll compute a y value and then we'll estimate the value of a limit so our function is this polynomial f of x is minus 7x squared plus 13x minus 25. Question A is to find f of 2. Well the solution is you simply substitute the x value negative 2 into this expression everywhere there was an x.
So you end up with this expression. When you compute that expression using the standard rules of arithmetic you get the result negative 79. So we would say that f of negative 2 is negative 79. That's a y value on the graph of f at x equals minus 2. Let's go on. Question b says use a table of xy values to estimate the value of this limit, the limit as x approaches negative 2 of f of x. Well that symbol is telling us that we need to consider what happens to the values of f of x when x gets closer and closer to negative 2 but not equal to f of x.
to negative 2. We can experiment by making a table of x and f of x values. Notice in the left column we put values of x that are getting closer and closer to negative 2 but not equal to negative 2. So minus 2.1, minus 2.01, minus 2.001 and so forth. In the right column we put the resulting values of f of x, computing using that formula and computed using a counter. calculator.
So you can see that this is fairly straightforward. When you substitute these x values into the formula for f, you get these expressions, you evaluate those using a calculator, and you get this column of results. Now remember that limits are about trends. In this table we see a trend in the x values.
These x values are getting closer and closer to negative 2. Now let's see if we can discern a trend in the y values. Well, the y values seem to be getting closer and closer to negative 79 as we go down that right column. So it looks like the values of f of x are getting closer and closer to negative 79. Now the table that we just built had values of x that are getting closer and closer to negative 2 and not equal to negative 2, but they also had the property that the x values are all less than negative 2. If we go up and look at those, we can see that all of those x values are less than negative 2. We should also build a table with values of x that are getting closer and closer closer to negative 2, but always greater than negative 2. That's what I've done in the table shown here. Notice in the left column, I have x values that have the trend that they're getting closer and closer to negative 2, as in the previous table, but this time they're all greater than negative 2. Now let's see if we can discern a trend in the values of f of x in the right column.
Well notice, those y values, those values of f of x, are getting closer and closer to negative 79. Based on these two tables we could write the following observation. When x gets closer and closer to negative 2 but not equal to negative 2 we would estimate that the value of f gets closer and closer to negative 79. Now we could abbreviate that whole sentence with limit notation. That would be this.
We would estimate that the value of this limit is negative 79. That's the end of our first example. Now I have some remarks about this first example. First of all the comparison of the limit and the y value. Notice that our estimate of the value of the limit that we found in part B, negative 79, matches the value of the limit. the y value that we found in part a, negative 79. A natural question is this, does the value of the limit as x approaches c of f of x always match the value of f of c?
Well, the answer to that is to think back to limits video A. In that limit, we saw examples of a function given by a graph where the value of the limit did not match the y value. In this first example that we've just done, it happens that the value of the limit does match the y value, but we should not expect that this will always happen.
A second remark that I want to make about this example is that it's very unsatisfying. In question, In question B, we had to use a calculator to find the values of f to fill those large tables. This kind of stuff could be done by hand if you took a lot of time, but it would really be a frustrating experience.
And we could only estimate that the values of f are getting closer and closer to negative 79. So we could only estimate that the value of the limit was negative 79. So the question arises then, is there a better way? is there some way to analyze the formula for f of x to determine the value of the limit precisely without estimating? Well, the answer is there are analytical techniques that are developed in a higher level math that do provide a way of analyzing the formulas for certain kinds of functions to determine their limits.
That's the good news. The bad news is that the techniques themselves, the analytical techniques, are beyond the level of an introductory calculus course. But the good news is that the math is not as good as the mathematical techniques.
is the general results of using those techniques can be presented as theorems that can be used in our course. Three such theorems about limits are presented on the next two pages. So here's our first theorem about limits.
It's properties of limits. If f and g are two functions and if they have these properties that the limit as x approaches c of f of x equals l and the limit as x approaches is C of g of x equals m where L and M are both real numbers that is to say both of these limits exist if all that stuff is true then all of this stuff is true all of these properties of limits so for instance if I consider the function obtained by adding the functions f of x and g of x and I take the limit of that function well then I get the same result that I would get by taking the limit of f separately and the limit of g separately and adding those limits together. So that's kind of what you expect, that is the limit of a sum of functions, that's this quantity, is equal to the sum of the limits of the functions, that's this quantity, the green quantity on the right. Now keep in mind that this property of limits that I just discussed, property 3 is only valid if f and g satisfy these properties that I discussed earlier. The limit of f of x exists and the limit of g of x exists.
Another property of limits is on line 5, suppose you take that function f and you multiply it by a constant k to get a new function called k f of x. And suppose you take the limit of that function. Well, it turns out that you get sort of what you you would expect. That is, you get the same value that you would get if you just take the limit of f by itself and then multiply that limit result by the number k. So in other words, if the limit of f by itself is L, then the limit of k f is the number k times L.
Now remember that property is only valid if f satisfies this, if the limit of f by itself does exist. Now there are There are some simpler properties in this table. This one that says the limit of a constant k is just that constant k.
That is the limit as x approaches c of the constant k is equal to the constant k. This property number two is also fairly simple. The limit as x approaches c of x is just c. Let's go on. The next theorem we'll discuss is this theorem three.
It's more of properties of limits. It's about limits of polynomial functions and rational functions. It says this, for a polynomial function f, the limit as x approaches c of f of x equals f of c.
So that gives us a way of computing the limit easily for a polynomial function. If we wanna compute the limit, we just simply compute the y value and they'll match. Now that brings to mind two earlier questions.
Is there some way to analyze the formula to find the value of the limit precisely without estimating? Well, yeah, that's one of the things we just found in that theorem. And that also brings to mind the earlier question that we asked. Does the value of the limit always match the y-value of the function? Well, the answer to that was not always.
But we see that what's significant about polynomial functions is that for polynomial functions the values of the limits will always equal the y values of the function. Another property listed in this theorem 3 is this one. For a rational function r with a nonzero denominator at x equals c, if you want to compute the value of the limit as x approaches c of r, you can compute that limit by just simply computing the y value r. Now so far, theorem 2 and theorem 3 have been about cases where limits do exist. This limit of a polynomial will exist because it's this y value for a polynomial.
And polynomials have domain, the set of all real numbers, so y values will always exist. This limit of a rational function will exist because it equals this y value for that rational function. And that rational function has...
has already passed the test that its denominator is not 0 where x equals c. So remember that rational functions have a domain, the set of all numbers that do not cause the denominator to be 0. That is, if you use an x value that does not cause the denominator to be 0, then you will get a y value. So this limit of a rational function that passes that test will always exist. In this previous page, this theorem was... all about situations where a limit would exist.
But remember from our previous video about a limit for functions given by graphs that there are plenty of situations where a limit does not exist. Well, it's going to turn out that for functions given by formulas, there are also going to be situations where a limit does not exist. Theorem 4 is about that.
Theorem 4 is about limits of quotients in certain situations. If you have the limit as x approaches the limit, approaches c of the quotient f over g, and if the limit of the numerator alone is a number l that's not zero, and the limit of the denominator alone is zero, then in that situation the limit of the quotient does not exist. In this video we'll do three examples that use theorems 2, 3, and 4. In our first example we'll revisit example 1, this time instead of guessing or estimating the value of the denominator, we'll use the derivative of the denominator.
I'll use a limit we'll find it precisely using a theorem. So this is again example one continued. Question C is to use the theorems about limits to find the value of the limit as x approaches negative 2 of f. Well you can see that my solution is here on one line.
I start with the quantity that we've been asked to find. I write that here on the left. The limit as x approaches negative 2 of f.
And then I just simply expand that, that is I replace f with its formula. So instead of the symbol f of x here, I have the formula for f of x here. Now kind of zoom out and look at the overall form of what we have. What we have there is a polynomial function with a limit expression parked in front of it.
So we're being asked to compute the limit of a polynomial. Well, if we go back up and look at theorem theorem 3 says for any polynomial function f, if you are asked to compute the limit as x approaches c of f, you can compute that simply by finding f. So that is, you can replace this expression that involves a limit with this expression that just simply computes a y value. So in our case, we have this expression that involves a limit of a polynomial, and we can replace it with this expression. which is just the computation of a y value.
We substitute in x equals minus 2. That means we're computing a y value. So theorem 3 tells us that we can do that. And when we do this computation, we already know that we get the result negative 79. That is, we computed that y value a while ago in example 1a.
Now observe that we got the same number here, negative 79, that we estimated in part b. In part b, using these tables of values, that table and this table, we estimated that the value of the limit was negative 79. That was a lot of work. So we got the same answer here, but there was a lot less work involved.
In this time, we know that the result is correct. In part b, we just estimated. We weren't really positive that the limit had value negative 79. Now I want to say one more thing about this.
question C. Remember that I said we start by putting the quantity that we're being asked to compute on the left. And then notice that I have a series of equal signs. There's an equal sign there, there's an equal sign there, there's an equal sign here, and then I have this number, negative 79. So you could think of this as sort of like an accordion that's expanded.
You've got this quantity and you've got this quantity at the ends of the accordion. Now this whole sentence, this mathematical sentence, would read as follows. This limit equals this quantity, which equals this quantity, which equals this quantity. So that's one great big mathematical sentence.
Notice I even put a period at the end of this sentence. This equals this, which equals this, which equals this, period. So mathematical formulas have the same syntax.
and grammar as prose sentences. So you should be clear when you write things in mathematical symbols that what you're writing really is a sentence. It's a math it's a sentence abbreviated in mathematical symbols but it needs to be able to be read as a sentence all the same. That's the end of example 1. Let's go on.
In example 2 we have a function f given by this formula. f of x equals the square root of the quantity. 24 plus x squared.
Question A says to find f of 5. Well that just means we substitute x equals 5 into this expression everywhere there is an x and that's what I did here and I get the result of 7. Notice again this format for my layout of my solution. I start by writing the quantity that we're asked to compute and then I go stage by stage with quantities that are equal, related with equal signs. So this quantity equals this quantity, which equals this quantity, which equals this quantity, which equals 7. So that's one big sentence. f of 5 equals this square root expression, which equals this square root expression, which equals this square root expression, which equals 7, period. So that's one big sentence that you could read in words, or you could print it in mathematical symbols.
but it's the same sentence both ways. Question B says define the limit as x approaches 5 of f of x. Now remember that until we knew about those theorems about properties of limits we would only be able to estimate such a limit by making it table of x y values for that formula but using those properties of limits in in theorems two three and four we should be able to find this limit analytically and precisely so my solution format is the same as the previous solutions. I start by writing the quantity that we're supposed to find.
There's the limit. And then I simply replace the symbol f of x with the formula for f of x. So instead of the symbol f parentheses x, I have this radical expression.
Now we zoom out and think about the form of what we have. We have the limit of a radical expression. So in this expression the limit is outside the radical.
Well if we go back up and look at our properties of limits In theorem 2 part 8 that is I would call this theorem 2.8 we're told that if we have the limit as X approaches C of a radical expression where we have a function f of X sitting inside of a radical if we have that Well in certain situations we can move the limit expression from B being outside of the radical to being inside of the radical. There are situations where we can do this. They're described here. So that's our situation here.
We have a limit of a radical expression and the limit is outside the radical. Well theorem 2.8 tells us that we can move that limit expression inside the radical. So theorem 2.8 tells us we can replace this expression with this expression.
That's the left and the right. sides of theorem 2.8. Now let's focus on the thing inside the radical. Inside the radical we have this.
That's a limit of a polynomial. Now remember that theorem 3, this theorem, tells us that we can replace that limit expression with this expression. We can just simply substitute in x equals 5. So we end up with square root of 49 which is is the number 7. So notice again what I call the accordion form of this answer. We wrote what we were being asked to compute, and at the very end we found that the result was the number 7. So this giant sentence would be read, the limit as x approaches 5 of f of x equals this limit, which equals the square root of this limit, which equals this square root expression, which equals this square root expression, which equals this square root expression. 7. So in other words, the value of this limit is 7. Notice also in this solution, I have the limit expression included where it needs to be included.
So the original thing we were asked for had a limit as part of the question. Then we rewrote the expression, replacing the symbol for f of x with the formula for f of x. Well, we kept that.
that limit expression because all that we did in this first step was we replaced f of x with its formula. And in this next expression, there is also a limit symbol. That's because theorem 2.8 is about moving that limit from outside the radical to being inside the radical. Only in this step does the limit symbol go away. So you have to be careful of where you put the limit symbol and where you don't put the limit symbol.
Let's go on. Example 3. In this example, f is this messy function. This is what's called a rational function.
It's a ratio of polynomials. This is the standard form. This is what you call the factored form.
Question A says find f parentheses 1. That is, find f. Now most of you are probably more familiar with the standard form of rational functions, but it turns out that it's the factored form that's... often most useful when computing y values. That is, we're being asked to compute the y value f of 1. So here we go. We write down the expression we're being asked to compute, f of 1, and then we compute it.
We grab this factored form of the function, we bring it down and we replace all the x's with the number 1, and we end up with this, 0 over 8, which equals 0. Question b says, find the limit as x approaches 1 of f of x. Well again, you might be more familiar with the standard form of rational functions but it's the factored form that we'll use. So we're being asked to find this limit.
In the first step what I do is I write down the expression we're being asked to find. and then I simply replace the symbol f of x with the formula for f of x. That's the factored form of the formula for f of x. Now let's zoom out and think about the form of what we have. the limit of a rational function and we're finding the limit as x approaches 1 and for this rational function the number x equals 1 is in its domain because remember the domain of a rational function is all the numbers that do not cause the denominator to be 0 well we see that the number 1 x equals 1 will not cause the denominator to be 0 so x equals 1 is in the domain of this function Well, theorem 3 tells us if you have a rational function with a non-zero denominator at x equals c, and you're being asked to find the limit of that rational function as x approaches c, well, you can find that limit by just simply computing the y value r of c.
So we can replace this limit expression with this expression. That is, we can just simply substitute in x equals 1. So here's a limit. here's a y-value theorem 3 tells us that in this situation that limit is the same as that y-value well when we compute the y-value we again get 0 over 8 which is just 0 that's the same computation that we did up here so this is another example where we find a limit and it turns out that the value of the limit is just the same as the y-value question C says fine f of 3. Well we lay out our solution as I've done before. We write the quantity that we're being asked for, f parentheses 3. We're going to use the factored form of f of x.
So we go get the formula for f of x, we bring it down and we substitute in x equals 3 in all the spots where there was an x. And we end up with this, negative 4 over 0. Now that does not exist. There's no real number that equals that expression.
So negative 4 over 0 is not 0, it's not negative 4, it just simply does not exist. This symbol does not represent a real number. Question D says to find this limit, the limit as x approaches 3 of f of x. Well, this is the limit of a rational function.
This function f of x is a rational function. It has a numerator, it has a denominator. What if we took the limit of the numerator by itself.
The limit of the numerator would be this. The limit of this polynomial expression that's in the numerator. Theorem 3 says you can find the limit of a polynomial by simply substituting in the x-value. Well when I do that in the numerator I get the number negative 4. Notice that negative 4 is not 0. What if we took the limit of the denominator by itself?
So the limit of the denominator would be this limit. The limit of the quantity x minus 3 times the quantity x minus 5. So in other words, the limit of a polynomial. Theorem 3 tells us, again, we can find the limit of a polynomial by just simply substituting in the x value, and we get that the limit of the denominator by itself is 0. Well, let's go back up and look at theorem 4. Theorem 4 says if you are asked to find the limit of a quotient, and if it turns out that... that the limit of the numerator is some number that's not zero, while the limit of the denominator is zero, well then this limit does not exist.
So we have this expression that we were asked for, the limit of f of x, and then we have this. I just simply replaced f of x with the formula for f of x. And then we jump all the way to here.
We just simply write down that the limit does not exist. Notice. Notice that I did not write down an intermediate expression. In this case, we are being asked for the limit.
Theorem 3 says we could substitute in numbers and we get this which turns out to be zero. In the current example, we don't have a theorem that says we can substitute in numbers. We have this theorem that says if you're presented with this limit, then you just jump to here.
You say the limit does not exist. Where I did substitute numbers in was zero. earlier when I was investigating the numerator by itself and investigating the denominator by itself.
So in those cases I was able to substitute numbers in and actually get values for limits. I got a value of negative 4 for the limit of the numerator by itself and I got a value of 0 for the value the limit of the denominator by itself. So the limit rules, the properties of limits, allow you to plug in x values. only in situations where you're actually going to get a number that's the limit. This will come up more in coming videos.
Well that's the end of that example. Well that concludes limit video B. Thank you.