BAM!!! Mr. Tarrou. That was a quite jump. In this video we are going to continue our
discussion of limits. The basic idea of limits and how you find them in Calculus. So I just
did a video of finding limits both graphically and numerically. That is just looking at graphs
of functions and identifying limits, both the left and the right hand limits. If those
are equal then there is a two sided limit. And finding them with t-table. But both of
those techniques are very tedious. Now we are going to start looking at properties of
limits and learning how we can just use algebra, and not have to make t-tables and graphs,
to find limits. Now this video is going to mainly... I am going to run through the basic
properties. I am going to along the way some very basic examples that apply to each of
these individual properties. Then in the next video we are going to do lots of examples
that might be a bit more challenging. So this video is just the identities. If you just
want to see examples you want to watch my next video. Ok. Properties of Limits. Let's
not forget, if we are just learning, limits that we are finding the limit of a function.
We are letting x approach some 'a 'value on an open interval. And from this function,
the limit of this function as x approaches 'a' is equal to L. L is a y value, the response
variable, the dependent variable. What is the y value of the function approaching as
x is approaching the value of a? So keeping that in mind we are going to talk about the
two most core basic properties that are going to allow us to do everything else that is
in this video. All of the other properties are going to use these two. So I am going
to take time with these two and make sure I attempt to explain them the best I can for
you. Then hopefully that helps you for the rest of the video. So the limit of a constant
function. A constant function is f(x) equals c. And in Calculus now you should be used
to the function notation, but just temporarily let that f(x) be y and really force the idea
that you see that is y equals a constant value. Even way back in Algebra 1 years ago you started
learning that y equals a constant number is a horizontal line. So this blue line is a
specific example. It is y equals 2. This is an example of a constant function or f(x)
equals 2. Ok, and again what is a limit? We will be given some value of 'a' to allow x
to approach, and we are going to approaching that value of 'a' from both the left... denoted
by a negative exponent on 'a'... and from the right... denoted by a little plus sign
in the exponent of 'a'. And we want to approach that value from both sides and hopefully get
the same answer and have that two sided limit that we are looking for. A limit is a y value,
so when your line is a horizontal line... when you have a constant function it really
does not matter what 'a' is. No matter what x is going to approach, you are going to be
getting the same y value. That is the deal with a horizontal line or y equals a constant.
No matter what x is, the y value... the function value is going to be equal to whatever that
constant is. In this case with the drawing it is 2. So to the little identity or property
here, the limit as x approaches 'a' of c being a constant... The limit of c as x approaches
a is equal to c. Basically if you take the limit of a constant the answer is that constant
itself. The y value that the graph, that constant function has that horizontal line is going
to be the same everywhere and thus you don't even see 'a' being factored in here. The limit
of c as x approaches 'a' is c, it is constant, it is a flat line. The limit of the identity
function. Now the identity function is f(x) equals x, or again just temporarily some simple
algebra terms of y=x. So y equals x is a line with a slope of one and a y intercept of zero.
I have got it drawn here. Now the deal is no matter what the x is, the y is the same.
It says y equals x, f(x) equals x. So the answer is the x value. If you allow 'a' to
be here at 2, then as you approach a from the left and a from the right the y values
are going to approach the y value of 2. So the limit of x, that is the actually right
side of the equation... that is the function, the limit of x as x approaches 'a' is... as
x approaches 'a' but there is nothing here but x... So if x approaches 'a', this is approaching
'a' thus the limit of x as x approaches 'a' is 'a'. Again it does make sense if you go
back and you look at this function y=x, f(x)=x. Whatever x is approaching, in this case 2,
the y value is approaching the same thing which is also 2 here. So the limit of a constant
is the constant itself. The limit of the identity function, the limit of x as x approaches 'a'
is going to be that value of 'a'. These two basic properties are going to do, or help
us to work through all of these other basic properties that we are going to run through
the next probably 10 minutes. BAM! Ok, we got three properties up here. We have the
limit of the sum which says the limit as x approaches 'a' of f(x)+g(x) is equal to the
limit of each of those terms. The take the limit as x approaches a of f(x), we take the
limit of g(x) as x approaches a, and then add them. It says that right here. The difference,
if you can add two terms then you can subtract them. So the limit as x approaches a of f(x)-g(x)
is equal to the limit as x approaches a of f(x) minus the limit as x approaches a of
g(x). You subtract them as it says right here. So a couple of simple examples, again this
is just the properties and simple examples. The limit as x approaches -4 of x+9, a binomial.
It is straight line with a slope of one and a y intercept of 9. We could have done this
in 11.1 by drawing a picture and see the limit graphically. But with these properties and
knowing the property for finding the limit of a constant and finding the limit of an
identity function, lost it there for a second, I can rewrite this as the limit as x approaches
-4 of x plus the limit as x approaches -4 of 9. And this is the, keep saying parent
function, identity function and this is a constant. So the limit of x as x approaches
negative four, if you let x approach -4 then this is going to approach -4. And the constant,
you know, it is a constant horizontal line. The y value is always going to be 9, and the
final answer is 5. Don't forget again we are talking about limits. That is where the x
value is approaching 'a'. It is going to be getting infinitely close to 'a' and thus this
limit or the y value is also approaching negative four and... well this is going to equal 9
because that is a constant function or a horizontal line. The limit as x approaches 5 of 12 minus
x. Again a binomial, it could be a trinomial... It could have four terms but I did not want
to write that much... is the limit of the first term minus the limit of the second term.
The 12 is a constant. The x is now going to be when separated, the identity function.
So the limit of a constant is the constant itself. The limit of x is going to be whatever
the x value is approaching, whatever 'a' is. That is going to be 5. And 12-5 is equal to
7. WHOOO!!! Alright. Now we have got the limit of a product. Now on these two examples, especially
the bottom orange one, you are going to see that it is getting kind of ridiculous how
much work I am showing to do these problems. And you will also to notice that I am showing
a lot of notation here, and I am breaking these limits off to little pieces giving each
term their own limit notation, now I am going to give each factor its own limit. That is
why we are going to build into the last couple of properties which really is going to streamline
this process of finding a limit of functions. So the limit as x approaches 'a' of f(x) times
g(x) is equal to the limit as x approaches 'a' of f(x) times the limit as x approaches
'a' of g(x). Basically this is one term, it has got two factors... or is the product of
two numbers. We are going to find the limit of each of those factors and then multiply
them together. So the limit as x approaches 5 of -6x is equal to the limit as x approaches
5 of -6 times the limit as x approaches 5 of x. Basically again, if you understood...
hopefully I did a good enough job explaining the first two initial properties of the limit
of a constant and the limit of the identity function... what we are now seeing is that
we can pull apart addition, subtraction, multiplication, we will be able to pull apart division too.
That will be the last example. Basically write all into a bunch of tiny limits of constants
and identities and indeed here we get -30. To be, you know, extremely strict on this
idea of I can only find the limit of the identity function and I can only find the limit of
a constant, if I am asked to find the limit as x approaches -2 of 7x^2-4 well I need to
expand all of that multiplication out and the subtraction. I get 7 times x times x,
which is x squared, minus 4. Then pull apart with the limit of a product property and the
difference property, put the limit function on the 7. The limit as x approaches -2 of
7. Apply it to both of the x's as well. We are going to take the limit 7, the limit of
x, the limit of x, and the limit of 4 and use the product property and the sum or difference
property. I don't know why I wrote plus negative there. I should have just wrote minus the
limit of 4. But any way it is going to work out. Take the limit of each constant and each
identity function separately and get 24. But you might be going isn't -2 squared four,
and 4 times 7 is 28, and 28 minus 4 equals 24. Why am I showing all of this work. It
is just the small building blocks, the small baby steps to get to the bigger properties
which you are now starting to realize will have pretty obvious patterns. So let's talk
about the limit of a monomial and the limit of a polynomial. I love this stuff. So here
are those patterns. We have the limit of monomials. Don't forget with a monomial we are talking
about a single term. Basically a polynomial is a collection of monomials. These terms
have a real coefficient and an exponent which is a whole number. So like say 2x^2 is a monomial,
but 2 times the square root of x which is 2 x to the 1/2 power would not be considered
a monomial. So when you have a monomial as you saw in my previous examples, and it is
kind of the same with polynomials, you basically just plug them in. The limit as x approaches
'a' of b sub n times x to the nth is equal to b sub n times A to the nth. You can see
where the x is coming out and the 'a' is just getting plugged in as you saw in my last example
in the previous screen where I took it apart into all those tiny little steps. So the limit
as x approaches 4 of -2x^3 is equal to -2 times 4 cubed. That equals -2 times 64 which
is -128. And if you just have a collection of monomials it is the same thing. You just
take that value of 'a' and plug it in for x and find the limit of you polynomial. Just
one more time, let's remember that this looks just like straight substitution but it is
because we were able to build up on those small little building block properties. We
do not just always take 'a' and plug it into the math function we are dealing with. We
are not always able to get those limits. We are following some specific set of directions
here, properties, that allow us to do this. You might have to make that table or make
that graph to actually find the limit either visually or numerically with your t-table.
The limit of a power, the limit as x approaches 'a' of f(x)...your function... and then the
entire function is raised to a power of n. You are going to be allowed to take the limit
of the inside of that function. Once you find the limit, then apply the power of n. Such
as the limit as approaches 3 of -2x+11 squared. So that whole binomial is being squared. Well
I can just temporarily let the power of 2 float out here at the top, find the limit
as x approaches 3 of -2x+11, and then once we find that limit... this is a polynomial
again a collection of monomials which all have whole number exponents and real coefficients...
we go ok, that is a polynomial. I can just take the value of 3, plug it into that expression,
work it out and get 5. Then when I am done, apply that original power of two that was
on the binomial and say 5 squared is 25. I have two more properties. We have the property
of roots which is basically a rewritten version of finding the limit of a power which is finding
limits of roots. Then we have that last property which is how to find limits of quotients.
That is going to be where... when you go into the next video... where I do a lot more complicated
examples, if you will, most of our examples are going to be about finding limits of quotients
and limits of piecewise functions. Those are the ones that take the most amount of work
to find the limit. Like with quotients, you can having an issue where you are tying to
divide by zero. So here we have our last two properties. We have the limit of a root. We
know fractional exponents or rational exponents. We can basically rewrite this property in
terms of a rational exponent and thus relate it exactly to a power. But when you have your
radical notation, the limit of a root, the limit as x approaches a of the nth root of
a function and n does need to be greater than or equal to 2 for this to work. Why would
you take the first root. The first root is going to be the same thing. What is the fist
root of 4? 4. It does not make any sense. N needs to be greater than or equal to 2.
What you can do is, you can take the limit of the function that is inside that root just
like you could take the limit of the function that was inside your power or exponent. So
the limit as x approaches 'a' of the nth root of f(x) is equal to nth root of the limit
as x approaches 'a' of f(x). This is then equal to... go ahead and find the limit of
that function and then take the nth root of it. Such as over here. The limit as x approaches
3 of the square root of 4x^2-9. I picked this example by the way because at least in PreCalculus
a lot of my students when they see all of the values underneath say a square root, it
could be a cube root as well but whatever, when you see a bunch of values that are perfect
square roots in this example students will want to just say that this is 2x-3. If I cover
up that 9 and put one term inside this square root symbol, then the square root of 4x^2
is 2x. The square root of 4 and the square root of x squared. However once you get a
polynomial in there it does not work that way. So we cannot take the square root of
4x^2-9 as it sits. Algebraically that is going to be as simplified as it gets. We are going
to take the limit of that inside function. The limit as x approaches 3 of 4x^2-9. That
is a polynomial with real coefficients and whole number exponents. We are going to take
that three and plug it in. Work it out until we get a value of 27, then take the square
root of 27. The square root of 27. 27 has a perfect square in it. It is 9. It is 9 times
3. We can square root nine. The square root of nine is equal to 3. So this is our limit
in reduced form. Then the last property, I am not going to do any examples in this video.
Because really, that is going to be four of my six examples in my next video of actually
finding limits of some more complicated functions than what we had in this video. Four of them
are going to deal with quotients because you cannot divide by zero. So the limit as x approaches
a of f(x) over g(x) is equal to the limit of the numerator.. the polynomial in the numerator...
or the function excuse me... it may not be a polynomial... The limit as x approaches
a of the numerator divided by the limit as x approaches a of the denominator. We are
going to take those two limits and then divide them out. Of course that is not going to work
if the denominator comes out to be zero when we try to find that limit. Well some of these
quotients are going to be set up in such a way, or be possible where we can algebraically
manipulate them and make the fraction... the quotient... not change its value but appear
different. Manipulate it in such a way that we can take that value of 'a' and plug it
and not get an undefined value and thus find the limit by simply using properties instead
of doing again a graphical display of the function or a numerical t-table to find that
limit. So that is it for properties of limits. Let's do another video with six examples,
four of which are going to be dealing with this limit of a quotient. I hope you keep
watching. I am Mr. Tarrou. BAM! Go Do Your Homework:)