Limit of a Function using a Graph - Basic/Differential Calculus

Summary

In this educational video, Sir Kenneth of STEM introduces the concept of limits in calculus, explaining how to determine the limit of a function as x approaches a certain value. He illustrates the process with examples, including left and right limits, and discusses scenarios where limits do not exist due to differing values from either side. The video also covers how to interpret limits from a graph, providing a comprehensive understanding of this fundamental calculus topic.

Takeaways

• 📚 The definition of a limit in calculus is introduced, explaining how it represents the value a function approaches as the input gets closer to a certain point.
• 🔍 The concept of left and right limits is discussed, emphasizing that if they differ, the limit at that point does not exist.
• 📈 Examples are given to illustrate how to determine the limit of a function by evaluating the left and right limits separately.
• 📉 The script demonstrates that if the left and right limits of a function at a point are the same, the limit exists and is equal to that common value.
• 🚫 It is pointed out that if the left and right limits are not equal, the limit does not exist at that point.
• 📝 The importance of the limit of a function being a single value is stressed, and the notation for expressing limits is explained.
• 📚 The script uses the function f(x) = x + 3 to show that the limit as x approaches 2 is 5, illustrating a basic limit calculation.
• 📉 An example where the limit does not exist is provided, showing how the function behaves as x approaches zero from both sides.
• 📊 The script includes graphical examples to visually represent the concept of limits and how they can be determined from a graph.
• 📌 The process of finding limits from a graph is explained, showing how to identify the behavior of a function as it approaches specific points.
• 👨‍🏫 The presenter, Sir Kenneth of STEM, uses a teaching approach to guide the audience through the concepts of limits in a structured and educational manner.

Q & A

What is the definition of a limit in calculus?

The limit of a function f(x) as x approaches a constant 'a' is the value that the function approaches as x gets closer and closer to 'a', but not necessarily equal to 'a'. It is denoted as 'lim(x→a) f(x) = L', where 'L' is the limit.

What does it mean if the left and right limits of a function at a certain point are different?

If the left and right limits of a function at a certain point are different, it indicates that the function does not have a limit at that point, meaning the function's behavior is not consistent from both sides as it approaches that point.

How can you determine the limit of a function as x approaches a certain value using a graph?

You can determine the limit of a function from a graph by observing the behavior of the function as x gets closer to the specified value from both the left and right sides. If the function values converge to a specific number, that is the limit.

What is the difference between the limit of a function and the value of the function at a point?

The limit of a function as x approaches a certain point is the value the function approaches but does not necessarily reach, while the value of the function at a point is the actual output of the function when x is exactly at that point.

Why might the limit of a function not exist at a certain point?

The limit of a function might not exist at a certain point if the function's values do not approach a consistent number as x approaches that point, or if the function is undefined at that point.

What is an example of a function where the limit does not exist?

An example from the script is a function where as x approaches zero from the left, the function approaches negative one, and from the right, it approaches positive one. Since the left and right limits are not equal, the overall limit does not exist at x=0.

How does the concept of limits relate to the behavior of a function as it approaches infinity?

The concept of limits is used to describe the behavior of a function as it approaches infinity by considering what value the function approaches in the limit as x goes to positive or negative infinity.

What is the significance of the left and right limits in determining the overall limit of a function at a point?

The left and right limits are significant because they provide information about the function's behavior approaching a point from both directions. If they are equal, it supports the existence of a limit at that point; if not, the overall limit does not exist.

Can the limit of a function be infinity?

Yes, the limit of a function can be infinity. It indicates that as x approaches a certain value, the function's values increase without bound, approaching positive or negative infinity.

What does it mean when the limit of a function as x approaches a value is equal to the function's value at that point?

When the limit of a function as x approaches a value is equal to the function's value at that point, it means the function is continuous at that point, and there is no discontinuity or abrupt change in the function's value.

Outlines

00:00

📚 Introduction to Limits in Calculus

This paragraph introduces the concept of limits in calculus with a focus on the definition of a limit of a function. It explains that a limit is the value a function approaches as the input (x) gets closer to a certain point (a), and it may not necessarily be the value of the function at that point. The paragraph uses the notation \( \lim_{x \to a} f(x) = l \) to denote this concept. It also discusses the idea of left and right limits, emphasizing that if they are not equal, the limit does not exist. An example is provided to illustrate the calculation of the limit of a simple function, \( f(x) = x + 3 \), as x approaches 2.

05:01

🔍 Examples of Left and Right Limits

This paragraph delves deeper into the concept of left and right limits, providing examples to illustrate situations where the limit does not exist due to the left and right limits being different. It also presents a scenario where the left and right limits are the same, thus confirming the existence of the limit. The paragraph uses a graphical approach to demonstrate how to determine these limits, showing that the value of the function at a certain point may differ from its limit as x approaches that point.

10:04

📉 Analyzing Limits with Graphs

The focus of this paragraph is on using graphs to analyze and determine limits. It provides a step-by-step process for finding the left and right limits from a graph and discusses how to interpret the behavior of the function as it approaches certain values. The paragraph includes examples that demonstrate how to read the graph to find the limit from both the left and right sides, and it highlights cases where the limit does not exist due to the function's behavior on either side of the point in question.

15:05

📚 Conclusion and Further Exploration

In the final paragraph, the script wraps up the discussion on limits, summarizing the key points covered in the video. It invites viewers to explore more examples of limits and to continue learning about this fundamental concept in calculus. The paragraph reinforces the importance of understanding the process of finding left and right side limits and the conditions under which the overall limit exists or does not exist.

Keywords

💡Limit

In the context of the video, 'limit' refers to the value that a function approaches as the input (usually denoted by 'x') gets closer and closer to a certain point ('a'). The concept is fundamental to calculus and is used to define continuity and differentiability of functions. The script discusses the limit of a function as x approaches a constant 'a', and how it can be approached from the left or right, indicating one-sided limits.

💡Function

A 'function' is a mathematical relationship between two sets that assigns to each element from the first set exactly one element of the second set. In the video, the function is denoted by 'f(x)' and is used to illustrate the concept of limits. The script provides examples of functions and how their limits are calculated as x approaches certain values.

💡Open Interval

An 'open interval' is a set of all real numbers between two given numbers, not including the numbers themselves. In the script, it is mentioned that a function is defined on some open interval containing a certain number 'a', which is important for discussing the limit of the function at that point.

💡Continuity

Continuity in the video is implied when discussing the limit of a function. A function is continuous at a point if the limit of the function as x approaches that point is equal to the function's value at that point. The script explains how the limit can indicate whether a function is continuous or not.

💡One-Sided Limits

One-sided limits, also known as left and right limits, are the limits of a function as x approaches a certain point from the left or right. The script uses one-sided limits to illustrate scenarios where the limit of a function may not exist if the left and right limits are not equal.

💡Graph

A 'graph' in the video represents the visual representation of a function, where the x-axis represents the input values and the y-axis represents the output values. The script uses graphs to demonstrate how to find limits and to show the behavior of functions as x approaches certain points.

💡Approach

'Approach' in the script is used to describe the process of getting closer and closer to a certain value without necessarily reaching it. This is crucial in understanding limits, as it describes how x values get closer to the point 'a' without actually being equal to 'a'.

💡Existence of Limit

The 'existence of limit' is a condition where the limit of a function as x approaches a certain point is a definite number. The script explains how the existence of a limit can be determined and how it can fail if the left and right limits do not match.

💡Negative Infinity

In the script, 'negative infinity' is used to describe a situation where a function's values decrease without bound as x approaches a certain point from the left. This is an example of a limit that does not exist because the function does not approach a finite value.

💡Value

The term 'value' in the script refers to the numerical result of a function for a given input. It is used in the context of discussing the output of the function as x approaches a certain point and in determining the limit of the function.

💡Differentiability

Although not explicitly mentioned in the script, the concept of 'differentiability' is closely related to limits. A function is differentiable at a point if it has a derivative at that point, which is defined using limits. The script's discussion of limits indirectly touches on differentiability by showing how limits can indicate the behavior of a function at a point.

Highlights

Introduction to the concept of limits in calculus.

Definition of a limit as x approaches a constant value.

Explanation of the notation for limits in functions.

Description of when the limit of a function does not exist.

Illustration of left and right limits and their implications.

Example of calculating the limit of a linear function as x approaches a specific value.

Demonstration of how to determine the limit from both the left and right sides.

Example of a function where the limit does not exist due to differing left and right limits.

Use of a graph to analyze the behavior of a function as x approaches a certain value.

Explanation of how to find the limit of a function from a graph.

Example of a function where the left and right limits are equal, confirming the limit's existence.

Process of determining the limit of a function at a specific point using a graph.

Example of a function where the limit from the left is different from the function's value at that point.

Explanation of how to interpret the limit of a function as it approaches infinity.

Demonstration of a function's limit approaching negative infinity.

Clarification of when a function's limit does not exist due to infinite behavior.

Final summary of the process for determining limits from a function's graph.

Encouragement for viewers to explore more examples to understand limits better.

Transcripts

hello everyone i am sir kenneth of stem

teacher ph welcomes adding basic

calculus lecture

series and pagosa panetta and calls the

limit of

a function let's start with our

definition

unlimit dao we let f be a function

verifying given function defined on some

open interval

containing the number a marathon

constant a

except possibly at a itself

hindi

we can use other constant spreading c

padding a depending on signal gamma

in a reference then the limit of f our

function

as x approaches our constant

a i use a good nathan which is letter l

so we write everything here on our

definition as this is the limit of our

function

f of x as x approaches

a young volume

to one and only number l as x takes

values that are

closer and closer to a the path

[Music]

limits not in towards this value

we can say that our limit does not exist

on different areas as we approaches our

editor so let us have that as a note

the limit of a function refers to the

value

that the function approaches

just a function function

function value itself

input x and then we are going towards

our output now i

union function button limit

[Music]

so we read this as the limit of our

function

f of x value x naught n

i papal appetite k c and then the value

t take the nut and

y axis is equal to our l so that part we

read this as the limit of f of x

as x approaches c is equal to l

okay limit

towards two areas

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let us have our example we have the

limit of our function at

f x naught n x plus 3 this line this is

our f of

x and then this is equal to x plus 3

and then as our x approaches 2.

x naught and let me change the color x

naught n i approaching 2

x b

2 in that case

[Music]

the limit of our function x plus three

as our

x approaches two la la generation

negative theta s so behind it as

x approaches from the left

we have the limit of our function x plus

three

as x approaches two la la generating

us we are approaching from the right at

you

concerned

value as we approach from the left high

paraholdings are valiant and limit as we

approach from the right

quantity we are approaching two from the

left munna

[Music]

so we can say that the limit of our

function

x plus three as x approaches

two from the left a young value

nonviolent

that is just equal to five

tapas quantitative and mulana

the limit of our function x plus three

as our x approaches two from the left

side

the value of our limit as x approaches

two

no adding function x plus three is equal

to five

[Music]

now let us have our second example where

in value

left side right side limit wherein our

limit does not exist

so note if the limit of the function

from left and from right

are different kapakamake bacilla then

the limit does not exist

as we are approaching zero dito from the

left

value so we are getting closer

to negative one from the left little

casino

from the right nothing as we are

approaching zero due to six nothing

one pero positive since mckay by bayong

value as we approach from the left

we have here that is negative one and as

we approach from the right that is

positive one

then our limit as our x approaches zero

does not exist or d and e mathematics

of our limit does not exist now let us

have more examples

we have our function here so atom graph

nato this is our f of x

so hoka had an attention and limit

casual new una

the limit of f of x as x approaches two

hindi panatinto

approaches so let us get our left side

right side limit first so with that said

this is one this is two amponser nat and

night and sababa we have our the limit

of our function f of

x as x approaches 2 from the left

y that's also 2

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so we have the limit of our f of x

as x approaches two from the right

amount is still positive

we are still getting closer to so that

is our limit now left side

right side in that case kumai kita

nathan

they are both positive too then the

limit of our function

as x approaches two is just equal

to nothing that is nothing

happen ion value non-function

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that is equal to one so the value of our

function f of two is equal to one

while our limit no adding function is

equal to

positive two do not unlimit at function

now let us try this one the limit of our

function f of x as x approaches three

measurements on partners if we look at

our x we have one

two and then three young values

straight line as we approach from the

left

we are approaching at y is equal to 2.

the limit of our function f of x as

x approaches 3 from the left is equal to

hindi negative

value value that is still equal to

positive 2.

where the value of our limit still

nugating function

as x approaches three pair from the

right palapitnamantayo

that is still 2. so the value of our

limit of our function

as x approaches 3 is still equal to the

help of

your left

[Music]

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so this is the process

in a limit chakanan function given a

graph so let us have one more example

so we will be using this graph of our

function to find each of the following

so the limit of our function had been

given up in a graph as

x approaches two at the unconcerned

from the left negative

we are also getting closer to the value

of y naught and that

is equal to positive one so as we are

approaching from the left

the value of our limit as x approaches

two that is equal

to positive one then for our next one

we are approaching

hindi

[Music]

so the value of our limit now adding

function as we approach

two from the right is equal to negative

six

taps

left side character right side limit

make a c1 at negative six

then our limit num function as

x approaches to mismo does not exist

and then the value of our function f of

two platform

and r2

[Music]

so the limit of our function as we

approach two from the left is one

from the right is negative six your

limit num function itself does not exist

periunvoluntal function that is equal to

positive four

if you want to do more examples let us

proceed in saving attention

so we have here problem number two

value i'm concerning as we approach one

from the left from the right value

function at one so i'm concerned nothing

red dotted

we are getting closer and closer and

closer kumai kita nathan

pava balangaba

[Music]

towards the negative y axis we are going

towards the negative

infinity as a totalitarian

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no adding function as we approach one

from the right case

so with this one we can say that the

value of our limit

is just equal to the value of y here

that is negative two

and then again from the left we are

going downwards towards negative

infinity

we are just approaching negative 2 then

our limit does not

exist and then the value of our

functions

as a graph it is at negative eight so

our function is

at negative eight statistic

if our value adding white as we approach

a certain value

[Music]

negative infinity so that is for problem

number two and then last one for problem

number three

we are approaching zero from the left

from the right

volume

volume limit as x approaches 0.

0 y-axis so as you approach from the

left

[Music]

0.

left side at right side limit nothing

then they are both one the value of our

limit

as x approaches zero is equal

so this is the whole process

your left side right side limits are in

value in the limit itself

function graph i hope you have learned

something thank you for

watching

[Music]

you