Limits in Calculus: Definition & Meaning. What is a Limit?
Summary
TLDRThis script offers a comprehensive explanation of the concept of limits in calculus. It begins with a layman's approach, explaining how limits work for smooth, continuous functions by plugging in the value of 'a'. The script then transitions to a more rigorous mathematical definition, introducing the symbols 'Delta' and 'Epsilon' to represent infinitesimally small distances. It visually describes how as 'X' approaches 'a', the function 'f(x)' approaches 'L', using graphical illustrations to clarify. The script concludes by demystifying the formal mathematical definition found in textbooks, making the abstract concept of limits more tangible.
Takeaways
- π The limit of a function f(x) as x approaches a number a is defined as a value L, which the function gets arbitrarily close to without actually reaching it.
- π In layman's terms, a limit is the value a function approaches as the input gets infinitely close to a certain point, without the input actually being that point.
- π Mathematically, limits are described using the concept of 'delta' (Ξ) and 'epsilon' (Ξ΅), which are small numbers representing the proximity of x to a and f(x) to L, respectively.
- π The definition of a limit is rigorous and involves the idea that for any small distance from a (denoted by Ξ), there exists a corresponding small distance from L (denoted by Ξ΅) for the function's value.
- π The concept of approaching a limit can be visualized graphically, where x values close to a result in f(x) values close to L, regardless of approaching from the left or right.
- π The definition of a limit is foundational in calculus and is used to prove the existence of limits under certain conditions, which is essential for understanding more complex calculus concepts.
- π’ The mathematical framework for limits involves the idea that if the distance between x and a can be made arbitrarily small (i.e., less than any given positive Ξ), then the distance between f(x) and L can also be made arbitrarily small (i.e., less than any given positive Ξ΅).
- π The script emphasizes that understanding the formal definition of a limit is crucial for reading and comprehending more advanced mathematical theorems and definitions in calculus.
- π― The practical application of limits in calculus often involves using limit laws or rules to solve problems, which builds upon the foundational understanding of what a limit represents.
- π The video script serves as an educational resource to demystify the concept of limits for students, providing both a high-level understanding and the detailed mathematical framework necessary for deeper study.
Q & A
What is the basic concept of a limit in mathematics?
-The basic concept of a limit in mathematics is that the value of a function approaches a certain value L as the input x approaches a certain value a, even though the function may not be defined at x = a itself.
Why is it important to approach a limit without actually reaching the value 'a'?
-Approaching a limit without reaching the value 'a' is important because it allows for the consideration of the behavior of the function around the point 'a', including cases where the function may not be defined at 'a', ensuring that the limit can be discussed in a broader context.
What is the role of the small number 'Delta' in the definition of a limit?
-The small number 'Delta' in the definition of a limit represents a range of values around 'a' that x can approach. It is used to define how close x can get to 'a' without actually being equal to 'a', thus allowing for the concept of infinitesimally close values.
How does the value 'Epsilon' relate to the definition of a limit?
-The value 'Epsilon' (Ξ΅) is used to measure how close the function's value can get to the limit L. As 'Delta' becomes smaller, 'Epsilon' also becomes smaller, indicating that the function's value gets arbitrarily close to L as x gets close to 'a'.
What does it mean to say that a function is 'smooth and continuous'?
-A function is 'smooth and continuous' if it has no breaks, jumps, or cusps. It implies that the function can be drawn without lifting the pen from the paper, and it is differentiable within the interval being considered.
Why is it necessary to use absolute values in the mathematical definition of a limit?
-Absolute values are used in the definition of a limit to ensure that the approach from both sides of 'a' (from the left and the right) is accounted for. It disregards the direction of approach, focusing only on the magnitude of the difference between the function's value and the limit.
How does the concept of a limit relate to the graphical representation of a function?
-In the graphical representation, a limit is shown as the value that the function's graph approaches as the input x gets infinitely close to 'a'. This is visualized as the graph getting closer and closer to a horizontal line at the value L, without necessarily touching it at x = a.
What is the significance of the phrase 'for every positive number Epsilon' in the definition of a limit?
-The phrase 'for every positive number Epsilon' signifies that no matter how small a positive distance from L one chooses, there exists a correspondingly small distance from 'a' (measured by Delta) such that the function's value will be within that small distance of L.
Can you provide an example of a function that is not smooth and continuous?
-An example of a function that is not smooth and continuous is a step function, such as the Heaviside function, which has a sudden jump at a certain point, or a function with a cusp, like f(x) = |x| at x = 0.
What are some practical implications of understanding limits in calculus?
-Understanding limits is fundamental in calculus for evaluating the behavior of functions, determining slopes of tangents, computing areas under curves, and understanding the convergence of infinite series, among other applications.
Outlines
π Introduction to Limits
The paragraph introduces the concept of limits in calculus. It explains that the limit of a function f(x) as x approaches a certain value 'a' is a value 'L', which the function gets infinitely close to but never actually reaches. The explanation is given in layman's terms and then transitions into a more mathematically rigorous approach. The speaker emphasizes the need to visualize this concept through a graph and prepares to draw one to further illustrate the definition of a limit.
π Graphical Representation of Limits
This paragraph delves into the graphical representation of limits. A function is sketched on a graph, and the concept of approaching a certain value 'a' from both the left and right is introduced. The speaker uses the terms 'a minus Delta' and 'a plus Delta' to represent values close to 'a' but not equal to it. The idea is to approach 'a' without ever reaching it, which is a key aspect of the limit definition. The paragraph also introduces 'Epsilon' as a small number that represents the difference between the function's value and the limit 'L' as 'x' approaches 'a'.
π Deep Dive into the Limit Definition
The paragraph provides a deeper exploration of the limit definition. It explains that for the limit to exist, for every small 'Epsilon' (a positive number representing the allowed error), there must be a corresponding small 'Delta' (a positive number representing how close 'x' can be to 'a') such that if 'x' is within 'Delta' of 'a', the function 'f(x)' is within 'Epsilon' of 'L'. The speaker clarifies that 'Delta' and 'Epsilon' are mechanisms to describe the infinitely close approach to 'a' and 'L', respectively. The paragraph also emphasizes the importance of understanding this definition to grasp more complex mathematical concepts.
π Textbook Definition and Practical Understanding
The final paragraph contrasts the graphical and conceptual understanding of limits with the formal definition found in textbooks. It reiterates that the limit of a function 'f(x)' as 'x' approaches 'a' is 'L' if for every 'Epsilon' greater than zero, there exists a 'Delta' such that 'x' being within 'Delta' of 'a' results in 'f(x)' being within 'Epsilon' of 'L'. The speaker acknowledges the complexity of this definition and encourages students to understand it as it is foundational for grasping other theorems and concepts in calculus. The paragraph concludes with an invitation to learn more about limit laws, which will aid in solving different types of limit problems.
Mindmap
Keywords
π‘Limit
π‘Continuous function
π‘Delta (Ξ)
π‘Epsilon (Ξ΅)
π‘Approaching a value
π‘Graphical representation
π‘Mathematical rigor
π‘Definition of a limit
π‘Absolute values
π‘Limit laws or rules
Highlights
The limit of a function as X approaches a certain number 'a' is defined as the value 'L' that the function approaches.
For a smooth and continuous function, the limit can be found by simply substituting the value of 'a' into the function.
Mathematically, a limit is defined by getting infinitely close to a certain value 'a', causing the function to approach a value 'L'.
A graphical representation of a limit involves drawing a function and approaching a point 'a' on the x-axis.
The value 'L' is the y-coordinate where the function intersects the vertical line at x = 'a'.
The concept of approaching a limit involves considering values close to 'a' but never actually reaching 'a'.
The Greek letter 'Delta' (Ξ) is introduced to represent a small number that the function's x-values can approach but not reach.
The function's values at points close to 'a' are labeled as 'L' plus or minus a small number 'Epsilon' (Ξ΅).
As 'Delta' becomes infinitely small, the function values approach 'L', with 'Epsilon' also becoming infinitely small.
The definition of a limit is a mathematical proof that requires a framework to show the limit's existence for any function.
The definition involves the function being defined in an interval containing 'a', except possibly at 'a' itself.
The formal definition of a limit involves 'Epsilon' and 'Delta', where for every positive 'Epsilon', there exists a 'Delta' such that if the absolute difference between X and 'a' is less than 'Delta', then the absolute difference between the function and 'L' is less than 'Epsilon'.
The absolute value is used in the definition to account for approaching 'a' from the left or right.
The definition of a limit is fundamental for understanding calculus theorems and proofs.
Understanding the definition of a limit helps in interpreting more complex mathematical concepts and theorems.
The limit definition is not just theoretical but also practical for solving limits in calculus.
Transcripts
we say that the limit
as X approaches some number a of some
function f of x is equal to some limit l
so basically if it's a smooth continuous
function with no breaks or anything you
just plug the value of a in there and
that's what the limit is now
mathematically speaking what you're
doing is you're getting closer and
closer and closer and closer to this
value a and the function because of that
is getting closer and closer and closer
and closer to the value L now that's
layman's terms that's Json teaching you
what it means but mathematically you
have to be a lot more rigorous so let me
draw a picture of what this definition
of a limit is and then I'm going to
slide this board back where I have
written the definition underneath and
then we'll pick it apart so you can
understand what your book's trying to
tell you
all right the bottom line is let's draw
a graph and I'll try to make it pretty
big
all right just so we can kind of draw
some things here so this is X and this
would be f of x so it's just any
function now when we do definition of a
limit it applies to any function f of x
so let's just draw some random functions
so we'll do this
something like this who knows what this
function is but it's some function of X
notice that this particular one is
smooth and continuous and we're just
going to do that for now but just keep
in mind that for the limit to really
exists at a point we'll talk about more
exactly what what conditions limit
exists but for now we're just going to
say this guy's smooth and continuous
because it's easier to understand
and then we say that
notice we said here as X approaches some
number a so for for the sake of argument
let's say that a is right here so we're
trying to find as X approaches the value
a this could be 3 like X approaches 3 x
approaches 4 x approaches five whatever
it's any value a as X approaches some
value a so I'm going to actually go
ahead and do a dotted line up here and
let's see what the function actually
says so here's where the function is at
that value x equal a and if you go
across over here you can see that the
corresponding value of a function is
listed right over here
okay and this value right here is what
we call L
okay so what we're basically saying in a
in a picture is the limit as X
approaches this value of a of this
function is equal to l it should be
totally clear what we're trying to say
because we've been doing this over and
over and over again
all right now when we define the
definition of the limit we have to
introduce some additional ideas because
you got to remember the definition of
this limit is a mathematical proof you
see I've already told you that limits
exist under certain conditions I've
already told you you get infinitely
close to it I've already told you all
these things but really in math you
can't just like tell somebody in
layman's terms what it is you have to
construct some sort of framework some
sort of proof that shows that these
limits exist for every possible function
that we could draw on the board or at
least have some methodology that we
could then use to find any limit we want
okay so it's not enough just to draw
this here we have to have a few more
ideas
so we're going to introduce something
here so here is the X approaching a but
of course as you get closer and closer
and closer to a you're going to approach
that limit but let's introduce a value
close to a on this side and a value
close to a over here because as you know
you can approach the limit from this
direction approach the limit from this
direction you know as we were doing our
table of values we could plug in values
on either side of a and get closer and
closer and closer so we'll pick some
values close by and this value here
we'll call A Plus sum number Delta this
is a lowercase Delta you might have seen
the triangle as the uppercase Delta well
this is just a lowercase Greek letter
Delta okay and then this one is a minus
some number Delta do not worry about
what Delta means for now just know that
Delta is a small number
a teeny weeny number so I'm going to put
a little arrow here Delta is just a
small
number and I'll give you a little bit of
a preview why do you think we're doing
this a minus Delta and a plus Delta
because ultimately what we're really
trying to say when we Define a limit is
that you get closer and closer and
closer to the value of a without Ever
Getting to a that's the layman term way
of explaining it to you and now you know
that you've been knowing that for a long
time that mathematically what you do is
you say hey we want to get closer to a
so what we'll do is we'll pick a value
over here that's a plus a teeny weeny
number and we'll pick a value over here
that's a minus so starting with a minus
a teeny weeny number so this difference
here this value here between a and these
marks this is a tiny number called Delta
and ultimately we're going to let Delta
get really really really small which is
going to let us approach the value of a
and infinitely close to a that's the the
Delta thing is just a mechanism to allow
us to approach a and get very very close
to it without ever getting there and
you'll see this pop up in the definition
okay but anyhow if this is a very small
value to the right of a and this is a
very small value to the left of a what
do you think is going to happen
if I go up here what would what would
the value of the function be that would
correspond to this value of x because
this is just a little bit to the right
well I can of course go up to this guy
here and I see that it intersects the
function over there and I can go over
here and see where it intersects right
there okay so obviously it's
intersecting the F the axis here at a
value larger than l okay so what we do
is we label this l
plus some other small number Epsilon
this looks like a kind of a curved e
instead of you know like an e like this
you make a curve like a c and then you
put a little thing in the middle that is
a very small number called Epsilon so
we remember that now we have another
corresponding value very close to a but
on the left hand side we can do the same
kind of thing we can go up here we
intersect the function and we see where
the corresponding value is over here and
it intersects right here so then we can
draw a little line here and remind
ourselves as this is L minus some tiny
tiny number called Epsilon now I'm
drawing everything large on the board so
that you can see it but really this
value a plus Delta and a minus Delta
these two values they're really really
really really really really close to a
okay and because of that the
corresponding values of the function
that correspond to those are really
really really close to L one of them is
a little bit higher than L
um just a little bit higher by the small
number Epsilon
and this guy is just a little bit lower
than L just a little bit lower by a
small number called Epsilon so again
this number here is a small
number the only real takeaway I want you
to get from this is that Delta is a very
small number in fact we're going to let
it become infinitely small as we
approach x x approaches a and we're
going to allow Epsilon because of the
way it's drawn here you can see that as
as these two guys get closer and closer
these are going to get closer and closer
so Epsilon becomes a very very small
small number as well
so the takeaway that I want you to to
see from here okay is something I'm
going to write on the board here and
it's something that you can see from the
drawing
but I want to make sure to write it down
okay if
if we make
if we make
Delta vary
small
what is that going to do what's going to
basically happen is these points if we
make Delta really really small these
points are going to get closer and
closer and closer and closer to a that's
what's going to happen what we're going
to say is that X is going to approach a
so this is the mechanism that allows us
to take our test value as we plug it in
our table or whatever and it gets
infinitely close to a making Delta small
is just basically the mechanism that
does that and then I'm going to draw a
little arrow and then I'm going to say
then
and I'll switch colors just to break it
up as a consequence of making this guy
get closer and closer to a
what happens then is that Epsilon
becomes
very small
and what that means is that f of x
approaches the limit l
okay so ultimately what I've drawn on
the board really is the entire
definition of a limit but it's done
graphically and it's done without a lot
of words but it's just a concept and
that concept is saying we have some
function here the red line okay we want
to approach a we want to get infinitely
close to a and you can see from the
drawing that as we get infinitely close
to a what's going to happen is the
function is going to get infinitely
close to the to the Limit L and we say
that in math that the limit as X
approaches a of the function f of x is
equal to l and that's a hundred percent
true okay but the way that we actually
pull it off mechanically in the
framework of mathematics is we have to
set a framework up and what we do is we
say okay we want to get closer and
closer to this so we'll pick a value to
the right of this and we'll pick a value
to the left of this and the value only
differs by this tiny tiny number that we
call Delta both of these points are
equidistant on this side to the left and
to the right by a tiny tiny thing Delta
okay now as the drawing stands these
correspond to value values up here one
of which is a little bit higher and a
little bit lower than the limit that
we're seeking L but what we're going to
do is we're going to make Delta get very
very small so what's going to happen is
we're going to encroach closer and
closer and closer and get infinitely
close to a and as we do that you can
trace your little fingers up and see
that the limit is going to approach L
because Epsilon is going to get smaller
and smaller as you do that that is
essentially the definition of the limit
if you understand that you can probably
stop the video now and just say that you
at least understand what's going on but
let me flip the board back and show you
what you're going to read in a typical
textbook
okay so remember that this is what the
definition of a limit is this is what
you're going to read in a textbook when
most student students read this they're
absolutely
um I mean I know I was dumbfounded when
I read I was like what does this
actually mean I think I understand
limits but I don't understand this so
this is the definition of a limit let f
of x be a function defined at each point
in some interval containing a except
possibly a itself
all right so the top part of the
definition if I want to break it down
for you all it says is let f of x be a
function defined on an interval that
contains some number a sometimes you may
it may not contain the number a just
depending on how your function is
defined okay because you can have
piecewise defined functions in certain
intervals but for the sake of
understanding this just the black text
just tell yourself okay all this is
saying is that their function exists
there's a point a inside an interval
that's basically all it says the
definition of the limit is down here
below it says then a number L is the
limit of f of x as X approaches a in
terms of this terminology which we've
been writing down this whole time okay
now here's the kicker here's the punch
line here's the part of it that that you
you know you wouldn't understand without
the drawing before so this is the limit
as X approaches a of the function
becomes L if for every number Epsilon
that's a positive number Epsilon is
greater than zero the there is a number
Delta that's also a positive number that
satisfies these conditions what this
first one says is if the difference
between X and A is between 0 and Delta
then the difference between the function
and the limit is a very small number
Epsilon
you might have to read this over and
over and over to really understand what
it's saying but in words what's
happening is what it's saying is if I
let the if I let the difference between
X and A become really really small what
it's saying is the difference between X
and A gets really small because remember
Delta is a small number I told you that
Delta is an incredibly small number it's
like 80 80 small number that's what
Delta is so what I'm saying is if I let
my variable X the difference between X
and A is the distance you know I'm
approaching X right so if I'm tracing my
finger along the difference between X
and A if it becomes a really small
number between 0 and Delta then the
difference between the function and the
limit will be a very very small number
that's exactly what I told you 25 times
a second ago what I'm saying is as you
trace your fingers left and right every
point you have your fingers there's a
distance between the value you have and
a okay we do that as we make our tables
remember we plug in values farther away
and then closer and closer and closer
and closer right so the difference
between my finger and a okay if I let
that get really small if the difference
between X and A gets really small which
means this difference here is between 0
and an incredibly tiny number Delta all
that is saying is that my finger is very
very very very very very very very very
very very very close to a but it never
quite gets there then if I let this
happen then the difference between the
function and the limit L is also going
to be a really really small number okay
all it's saying is that as my finger
gets infinitely close to a then the
function gets infinitely close to L
because it differs the difference
between the function and L differs by a
very small number all it's saying is
that as I squeeze my fingers this way
then this becomes squeezed together as
well
the reason there's absolute values here
is because of the way I've drawn the
picture because I can approach from the
left or I can approach from the right
remember the absolute value all it does
is it just throws away the sign doesn't
matter if you're coming from the left or
if you're coming from the right it
throws away the sign so that's why those
absolute values are there because it
doesn't matter if I'm approaching from
the left of a or from the right of a it
doesn't matter if x minus a is positive
or if x minus a is negative I'm
approaching from the other way whatever
the difference is I just look at the
absolute value and it's going to be a
very small number because it's between 0
and Delta which is incredibly tiny okay
when this happens the function itself
approaches the difference between the
function and L approaches a very very
very small value or it's less than a
very very small value it doesn't matter
if I'm approaching from the top or the
bottom see this is f of x minus l
f of x minus L when if I'm approaching
from this way like if I'm coming this
way f of x minus L will be positive
because this number minus this number is
positive but if I'm coming this way then
this number f of x minus L will be
negative but you see the absolute value
throws it signs just throws it away so
the only reason we have absolute value
signs in that definition is because it
doesn't matter if you approach from the
underside from the left hand side to
find the limit or if you approach from
the right hand side to find the limit
okay that's the bottom that's the bottom
line is whenever we're looking at a
limit at a point like that okay then we
basically want to be able to come from
the left or from the right
and what's happening here is that if we
get infinitely close to this point a
which means the difference between the
point and a is very very small then the
difference between the function and the
limit approaches this very very very
small value there you have it okay that
is basically it we can let f of x get as
close as we want to L by taking X closer
and closer to a but never quite equaling
a now I told you that in the very first
section of what is in limit okay I mean
that's what that's what I'm here for so
trying to break it down for you that's
the bottom line I told you from the very
first day that you get closer and closer
and closer to L and then the function
gets closer and closer I should I'm
sorry let me start over you get closer
and closer and closer to a along the
axis and then the function approaches
closer and closer and closer to l
this is the mathematical framework that
does that we write the limit down like
this this business you really never use
it practically it's just defining what
the limit does it's just saying hey if
you get closer and closer and closer
then the limits can the function is
going to get closer and closer closer to
the Limit that's all it is saying and I
know that the first time I read that
definition of a limit I stared at these
and I stared at these and I stared at
these and I stared at these and I didn't
quite get it but the secret is knowing
that these two numbers are
infinitesimally tiny little numbers and
all it's saying is that if I approach
them on the axis here then the function
is going to approach that limit that's
all it's saying so there's no problems
for this section this is mostly just a
informational session for you so you
understand what this definition is and
as you move on to other sections you'll
understand in your back pocket formally
mathematically what it means it's also
good practice now that you understand
this to help you read these types of
theorems and definitions and calculus
because there'll be more as time goes on
and you'll kind of get a little practice
reading one of these guys you can
understand this one then you can
understand any theorem in a calculus
book because this is kind of one of the
more difficult ones to understand
all right so make sure you understand
this and then follow me on to the next
section we're going to uh talk about
something called limit laws or limit
rules and it's going to help you solve
different types of limits in calculus
learn anything at mathandscience.com
Browse More Related Video
Limits, L'HΓ΄pital's rule, and epsilon delta definitions | Chapter 7, Essence of calculus
Calculus - The limit of a function
Introduction to limits 2 | Limits | Precalculus | Khan Academy
LIMITE: a Ideia Fundamental do CΓ‘lculo
Jika lim x->-3 (x^2+4x+3)/(x+3)=a-1,nilai a adalah...
Limits of functions | Calculus
5.0 / 5 (0 votes)