Limits in Calculus: Definition & Meaning. What is a Limit?

Summary

This 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

• 📘 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.
• 📐 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.
• 📊 Mathematically, limits are described using the concept of 'delta' (Δ) and 'epsilon' (ε), which are small numbers representing the proximity of x to a and f(x) to L, respectively.
• 🔍 The definition of a limit is rigorous and involves the idea that for any small distance from a (denoted by Δ), there exists a corresponding small distance from L (denoted by ε) for the function's value.
• 📉 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.
• 📌 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.
• 🔢 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 Δ), then the distance between f(x) and L can also be made arbitrarily small (i.e., less than any given positive ε).
• 📝 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.
• 🎯 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.
• 📚 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' (ε) 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

00:00

📘 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.

05:00

📈 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'.

10:01

🔍 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.

15:01

📚 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.

Keywords

💡Limit

In the context of the video, 'limit' refers to the value that a function approaches as the input (x) gets closer and closer to a certain point (a). The video explains that mathematically, this is not just about plugging in the value of 'a' into the function, but rather about the function values getting arbitrarily close to a specific number 'L'. The concept is central to calculus and is used to define continuity and differentiability of functions. The script uses the phrase 'limit as X approaches a of the function f of x is equal to l' to illustrate this.

💡Continuous function

A continuous function is one that has no breaks or jumps in its graph. The video script mentions that the definition of a limit applies to any function, but for simplicity, it uses a smooth and continuous function to illustrate the concept. The idea is that as 'x' approaches 'a', the function values also approach 'L' without any abrupt changes, which is easier to visualize and understand in the context of limits.

💡Delta (Δ)

In the script, 'Delta' (Δ) symbolizes a small number that represents the distance by which 'x' can deviate from the point 'a'. The video uses 'a plus Delta' and 'a minus Delta' to demonstrate approaching 'a' from both sides. This concept is crucial for the formal definition of a limit, where Delta can be made arbitrarily small to show that the function values can be made arbitrarily close to 'L'.

💡Epsilon (ε)

Epsilon (ε) in the video is another small number that represents how close the function values are to the limit 'L' when 'x' is within a distance of Delta from 'a'. The script explains that as Delta gets smaller (approaching 'a'), Epsilon also gets smaller (approaching 'L'), which is the essence of the limit definition. The phrase 'a very small number called Epsilon' is used to describe this.

💡Approaching a value

The concept of 'approaching a value' is fundamental to the idea of limits. The video script uses the phrase 'as X approaches some number a' to describe the process of getting infinitely close to 'a' without actually reaching it. This is a key aspect of the limit definition, where the function values are said to approach 'L' as 'x' approaches 'a'.

💡Graphical representation

The video script includes a detailed description of drawing a graph to visualize the concept of limits. It explains how to plot a function and how the function values approach 'L' as 'x' approaches 'a'. This graphical representation is used to provide an intuitive understanding of the limit before delving into the mathematical rigor.

💡Mathematical rigor

The video emphasizes the need for mathematical rigor when defining a limit. It contrasts the informal, graphical understanding with the formal definition that involves precise conditions involving Delta and Epsilon. The script mentions that 'mathematically you have to be a lot more rigorous', highlighting the importance of formal definitions in mathematics.

💡Definition of a limit

The 'definition of a limit' is the formal mathematical statement that if for every positive Epsilon, there exists a corresponding Delta such that the function values are within Epsilon of 'L' whenever 'x' is within Delta of 'a'. The video script breaks down this definition to explain how it encapsulates the idea of approaching a value without reaching it.

💡Absolute values

The script mentions the use of absolute values in the formal definition of a limit to account for approaching 'a' from either side (left or right). It explains that the absolute value ensures that the difference between 'x' and 'a', and between the function value and 'L', is considered regardless of direction, which is crucial for the limit's definition.

💡Limit laws or rules

Towards the end of the script, the concept of 'limit laws or rules' is introduced as a tool for solving different types of limits in calculus. These rules provide a framework for calculating limits once the foundational concept is understood, which is a practical application of the theoretical definition discussed throughout the video.

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' (Δ) 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' (ε).

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