Properties of Limits Calculus 1 AB


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:)

68 Replies to “Properties of Limits Calculus 1 AB

  1. Thank you for watching:) I am glad I could help and you liked my videos so much. I hope you do great in your class…or studies:)

  2. Thank you for your videos, I have used countless of them and it has really helped me to understand and do better in math. I appreciate you taking the time to make these videos because you do a great job. My one suggestion is to never scrape your fingernails on the chalk board like you did in one of the algebra videos. I watched that video months ago but the sound can not be unheard and it still haunts me to this day. So again, thank you for videos but not for traumatizing me ๐Ÿ˜‰

  3. You are very welcome! Thank you for adding so many videos to your likes list, I appreciate the support:) But I guess I'll have to include a disclaimer regarding undue trauma may be caused to viewers by my chalkboards:(
    LOL…Seriously, THANKS for being such a loyal fan and I hope you will share my channel with your friends and classmates who might benefit from it:D

  4. You are very welcome! Thank you for adding so many videos to your likes list, I appreciate the support:) But I guess I'll have to include a disclaimer regarding undue trauma may be caused to viewers by my chalkboards:(
    LOL…Seriously, THANKS for being such a loyal fan and I hope you will share my channel with your friends and classmates who might benefit from it:D

  5. You are very welcome! Thank you for adding so many videos to your likes list, I appreciate the support:) But I guess I'll have to include a disclaimer regarding undue trauma may be caused to viewers by my chalkboards:(
    LOL…Seriously, THANKS for being such a loyal fan and I hope you will share my channel with your friends and classmates who might benefit from it:D

  6. You're welcome, glad I could clear that up for you:) And THANK YOU for liking and subscribing, I appreciate the support!

  7. And THANK YOU too for liking and subscribing…support like yours is how my channel groWs, so please share my channel with others too:D

  8. I'm taking Calc 1 and I'm sad that I won't be able to refer to your videos after this semester. ๐Ÿ™

  9. I too am sad to hear that:( But please remember where to send the others who will still benefit from my help:)
    I'm starting to work on my Calc library to prepare for my AP Calc class that I'll be teaching next year but I'm afraid that won't help you….thanks for choosing my channel for your Calc 1 though…and good luck with your future classes.

  10. Number 2 in my Calculus Playlist called: Finding Real limits Graphical & Numerical Approach!
    Hope I'm not too late with the info:(

  11. thank you sir and this video help me alot . you are the best teacher.the way you come to start the video brings enthusiasm to learn and a big smile on my face.and thank you sir for always replying me.

  12. You are sooo welcome!…and THANK YOU for taking the time to share such a great testimonial!
    I love to see and hear about students who take the extra step to supply yourselves with all the learning materials that are available out there for you. As long as you never "give up" and don't just "place blame" on someone or something else you will find that life's path will be much smoother:) Thanks for watching and supporting!

  13. I wish I would have had you as my teacher Mr.tarrou you have helped me so much these past 2 years…and you're still doing it right now . I'm in calc BC by the way and it's so easy to learn about limits when I watch your math videos …!!!

  14. WOW…then you have been around since the start of my channel…I just celebrated the completion of my second year on YouTube! I never dreamed when I started that I would be teaching student all over the word in over 150 countries:0 Thanks for being such a loyal viewer and I hope you are spreading the word for me too!

  15. Thank you, as a college student who only took precalc in high school, and jumped right into Calculus in college, you're a life saver. I'm so glad I can actually understand you're accent!!!! You will be the only reason I pass Calc I with a good grade.

  16. And THANKS for subscribing and choosing my channel to watch and learn from with my not so southern accent! Together we should be able to pass that class like BAM!!!
    We watched your video and I told my wife I was going to wear a v-neck shirt and no glasses and give a๏ปฟ try at learning to be a "hipster" for my next video…she said forget it, and reminded me that Huey Lewis said it's "Hip to be Square"…lol

  17. Hi! I'm in AP Calculus now and my teacher actually recommended your channel to us if we needed extra help. Just thought you'd enjoy that!

  18. I certainly DO!!! Mind if I ask who your teacher is and where you go to school? I love to hear how the word spreads and from how far…I recently had a new student that transferred to FL from Washington State tell me that she was already watching my videos when she lived there:) My wife and I are working very hard to get the word out about my๏ปฟ channel and always like to hear the details from stories like yours:) Thanks for sharing!!!

  19. Hey, thanks a lot for your videos, theyre amazing
    I just moved from the States to Germany, and their math is two years ahead ๐Ÿ™ so I took algebra 2 my junior year, but in germany, we're doing limits and derivatives, which is calculus, right… and i also have a math exam in two weeks…
    but my question is how do you determine if its a constant function or an identity function?

  20. y=x is the identity function and derivative is 1 and y=any number is a constant function which has a derivative of 0. It sounds like you have quite the adventure and challenge ahead of you. Thank you for your support:D

  21. You are so awesome for making these videos! ย Math has never been my strong point, but after watching these videos it just might be. ย I feel very confident in my calc class now that I actually know what's going on haha!! I can't say thank you enough for the time and effort you put into making these videos!!ย 

  22. This is the third term in a row I am coming to you for virtual tutoring :). Your videos are so easy to understand, and I love how you took the time to put them in order. Thank you so much for what you do!ย 

  23. Iam back its my first day in CSC 208 Iam out of the country and times zones are jacked up. I am glad that you do what you do. No way I would pass with out ya!!!!!!!! Must Subscribed those that are not sure!!!! College level any math Prof Rob Bob is that Man!!!

  24. Great video! You explain things very well. I've also made a few tutorial videos, mostly on limits at this point. I'm subscribing to your channel.

  25. I know you've heard this multiple times but it wont hurt for you to hear it again ๐Ÿ™‚ a million thanks for the videos, you explain everything so clearly! Awesome work!

  26. I've been watching your videos since high school, I still am watching them here years later at college. Thanks for the videos! They've gotten me far!

  27. Hello, love your videos, but I have a question about the answer you got for the limit of a root. Wouldn't there be two answers since when you square root something, you get a positive and negative answer?

  28. This isn't fair! Not fair at all! Why aren't you my math teacher? How come you are so good at teaching math while my teachers are not? You should definitely come to my school and explain my teachers how to teach. Your students are so lucky to have you and I envy them. Anyhow, Thank You SOOOO Much! Keep enlightening your students. Best of luck and Take Care!

  29. Limit Poetry. The product of the limit is the limit of the products. The sum of the limits is the limit of the sums. ETC..

  30. I am following an old syllabus and re-teaching myself Calculus instead of taking a $600 course and these videos are perfect for me. You break everything down and it's easy to follow which makes learning Calculus again on my own so much easier. Your enthusiasm and love for the subject really shows through and it keeps me interested and wanting to learn more!

  31. You are God's gift to everyone who wishes to explore and learn Math. Thank you very much! Hope you can upload more Math videos in the future. More power ๐Ÿ™‚

  32. Impeccable Sir!!!!! One of the amazing mathematics videos I have come through in youtube. these videos are amazing and very much helpful.

  33. For F*** sake, why does NOBODY show in the videos / articles about limits properties when you can or can't apply these properties?
    It's been half an hour that I'm jumping from video to video, trying to understand this.
    For example, so far I've understood that in the limit of a difference, the limits must be finite (to avoid subtracting plus infinity and minus infinity).

    You CAN'T do a video on this argument and not show these properties!

  34. Oh my goodness I just realized this is my SIXTH year coming back to you for math help! I'm taking AP Calc AB and I couldn't be happier to watch your videos which explain things so well!!!!
    Ross McKee
    North Carolina

  35. bam! (You're right it was a quiet jump) Is it ever possible to find the limit of an X value? I know that X is the independent and Y is the output of a function…. but can it be done to see what X is in terms of a Y? So as Y approaches a number, we can determine what the X value is? Would this be the same thing as an inverse function?

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