Calculus - The limit of a function

Summary

This video script explores the concept of limits in mathematics using a relatable pizza analogy. It explains how limits describe the value a function approaches, rather than its output, through the example of f(x) = 3x^2 - 1. The script clarifies that limits can differ from the function's value at a point, as demonstrated with the function (x^2 - 4) / (x - 2), which approaches 4 but results in an undefined expression when x equals 2. The explanation emphasizes the function's behavior leading up to a point, setting the stage for a deeper dive into epsilon and delta in a future video.

Takeaways

• 📚 Limits are a fundamental concept in calculus, allowing us to determine the value a function approaches as the input gets arbitrarily close to a certain point.
• 🍕 An analogy is used to explain limits, comparing the approach to a fridge with the approach of a function's value to a certain limit.
• 🔍 The script introduces the function f(x) = 3x^2 - 1 and demonstrates how to find the limit as x approaches 2, which is 11, by substituting values close to 2.
• 🤔 It highlights the difference between a limit and the actual output of a function at a certain point, emphasizing that limits predict approach rather than exact values.
• 🚫 The script points out that some functions have points where direct substitution is not possible, such as division by zero, yet limits can still be determined.
• 📉 The function (x^2 - 4) / (x - 2) is used to illustrate a limit where the function approaches 4 as x approaches 2, despite the function being undefined at x = 2.
• 🕳 The concept of a 'hole' in a function's graph is introduced, showing that a function may not reach a certain value but still have a defined limit approaching that value.
• 🔑 The importance of behavior leading up to a certain point is emphasized, indicating that limits describe the trend rather than the exact position on the graph.
• 📈 The script sets up for future content by mentioning 'epsilon and delta' definitions, which will provide a more precise explanation of limits.
• 👍 The video encourages viewer engagement by asking for likes and subscriptions, suggesting a community interested in learning more about limits.
• 🔗 Additional resources for learning about limits are offered, including further video examples and a lecture on the epsilon-delta definition of limits.

Q & A

What is the general concept of limits in mathematics?

The general concept of limits in mathematics is to determine the value that a function approaches as the input gets arbitrarily close to a particular point, without necessarily reaching that point.

How does the pizza analogy help explain the concept of limits?

The pizza analogy helps explain limits by comparing the approach to a destination (the fridge) with the approach to a value that a function takes as its input gets closer and closer to a specific number.

What is the function given as an example in the script to demonstrate limits?

The function given as an example in the script is f(x) = 3x^2 - 1.

What values were used to approximate the limit of the function f(x) = 3x^2 - 1 as x approaches 2?

The values used to approximate the limit were 1.9, 1.99, and 1.999.

What is the limit of the function f(x) = 3x^2 - 1 as x approaches 2, according to the example?

The limit of the function f(x) = 3x^2 - 1 as x approaches 2 is 11, based on the values obtained from the approximation.

Why can't we always find the limit of a function by simply plugging in the value?

We can't always find the limit of a function by simply plugging in the value because some functions are not defined at certain points, and limits focus on the value the function approaches, not necessarily the value it reaches.

What is the significance of the phrase 'arbitrarily close' in the context of limits?

The phrase 'arbitrarily close' in the context of limits signifies that we can get as close as desired to a certain value by choosing inputs that are sufficiently close to the point of interest, without actually reaching that point.

What is the second function used in the script to illustrate the concept of limits?

The second function used in the script is (x^2 - 4) / (x - 2).

What happens when you try to find the value of the function (x^2 - 4) / (x - 2) by plugging in x = 2?

When you try to find the value of the function (x^2 - 4) / (x - 2) by plugging in x = 2, you encounter an undefined expression, as it results in a division by zero.

What is the limit of the function (x^2 - 4) / (x - 2) as x approaches 2, despite the function being undefined at x = 2?

The limit of the function (x^2 - 4) / (x - 2) as x approaches 2 is 4, even though the function is undefined at x = 2, because the behavior of the function leading up to that point is consistent.

What mathematical concepts will be introduced in the next video to further explain limits?

In the next video, the concepts of epsilon and delta will be introduced to provide a precise definition of limits.

Where can viewers find more information about limits and related mathematical topics?

Viewers can find more information about limits and related mathematical topics on the speaker's website: MySecretMathTutor.com.

Outlines

00:00

📚 Understanding Limits with a Pizza Analogy

This paragraph introduces the concept of limits in mathematics through an analogy. It explains that limits are about determining the value a function approaches as the input gets arbitrarily close to a specific number, rather than the output of the function itself. The analogy of walking towards a fridge to satisfy a pizza craving is used to illustrate the idea of 'approaching' a value. The paragraph also introduces the function f(x) = 3x^2 - 1 and demonstrates how to find the limit of this function as x approaches 2 by using values close to 2, showing that the function approaches 11.

Keywords

💡Limits

Limits in the context of the video refer to a fundamental concept in calculus that describes the value that a function approaches as the input (usually denoted as 'x') approaches a particular point. The video uses the analogy of approaching a fridge to explain how limits work, emphasizing that limits are about the behavior of a function as it gets closer to a certain value, not necessarily the value of the function at that point. For example, the video discusses the limit of the function f(x) = 3x^2 - 1 as x approaches 2, which is 11.

💡Function

A function, in mathematics, is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The video script uses the function f(x) = 3x^2 - 1 to illustrate the concept of limits, showing how the output values of the function approach a certain number as the input values get closer to 2.

💡Arbitrarily Close

The term 'arbitrarily close' is used in the video to describe the concept that as x values get closer and closer to a certain number, the function's output can get as close as desired to a specific value without actually reaching it. It is a key aspect of limits, emphasizing precision and the potential for infinitesimally small distances.

💡Analogies

Analogies in the video serve as a teaching tool to explain the abstract concept of limits by relating it to a more tangible situation, such as walking towards a fridge when craving pizza. This helps the viewer to visualize and understand the idea of approaching a value without necessarily reaching it.

💡Output

In the context of functions, 'output' refers to the result produced by the function for a given input. The video clarifies that limits are not about the function's output at a specific point but rather the value it approaches as the input gets closer to that point.

💡Arithmetic Operations

Arithmetic operations such as addition, subtraction, and exponentiation are used in the video to define the function f(x) = 3x^2 - 1. These operations are fundamental to understanding how the function behaves and how its outputs change as the input values approach a certain number.

💡Epsilon and Delta

Epsilon and delta are introduced at the end of the video as terms that will be used to define limits more precisely in a future video. They are fundamental in formal definitions of limits, where epsilon represents a small positive distance, and delta is the corresponding small change in the input that ensures the function's output is within epsilon of a particular value.

💡Graph

The term 'graph' in the video refers to the visual representation of a function, where the x-values are plotted on the horizontal axis and the corresponding function outputs are on the vertical axis. The video mentions a 'hole' in the graph at x = 2 for the function (x^2 - 4) / (x - 2), illustrating that the function does not have a defined value at that point, yet the limit as x approaches 2 exists and is 4.

💡Undefined

The video script touches on the concept of 'undefined' in the context of a function not having a value at a certain point, such as when plugging in 2 into the function (x^2 - 4) / (x - 2), which results in a division by zero. This is contrasted with the limit, which can still exist even when the function is undefined at that point.

💡Behavior

The 'behavior' of a function in the video refers to how the function's output values change as the input values approach a certain number. This is central to understanding limits, as it focuses on the trend of the function rather than specific output values, as illustrated by the function approaching 4 despite being undefined at x = 2.

Highlights

Limits help determine the value a function approaches with a given input.

An analogy of approaching the fridge is used to explain limits.

The limit of a function is its behavior as it gets arbitrarily close to a value, not the output at that value.

Example given with the function f(x) = 3x^2 - 1 to illustrate how limits are approached.

Values obtained from inputs close to 2 suggest the function approaches 11.

Limits can be different from the function's output when the input value is plugged in.

Different scenarios with the pizza analogy demonstrate the concept of approaching a value without reaching it.

The function (x^2 - 4) / (x - 2) is used to show limits where the function does not reach the limit value.

Inputs close to 2 for the function suggest it approaches 4, despite a hole at x = 2.

The function's behavior leading up to x = 2 is consistent, indicating the limit is 4.

Limits focus on the behavior of a function and what it gets arbitrarily close to, not the exact value reached.

The concept of 'arbitrarily close' will be explained in the next video with epsilon and delta.

A call to action for viewers to like and subscribe to the channel for more content on limits and related topics.

An invitation to visit MySecretMathTutor.com for additional resources and videos.

Introduction of the next lecture video discussing the precise definition of a limit using epsilon and delta.

Transcripts

In a general sense limits allow us to determine what value a function is approaching when

we use a particular input.

Not necessarily what the function gives us as output, but rather what value its getting

arbitrarily close to.

Let's explain limits using an analogy.

Suppose you are watching T.V. and start getting a massive craving for pizza.

Fortunately you just happen to have some left over pizza in the kitchen.

So you get up from the couch and start heading towards the fridge.

In this instance if someone were to describe where you are going, they would say you are

approaching the fridge.

They would be confident in this description because as you keep walking, you are getting

closer and closer to where the fidge is located.

This example is the same thing we want to do with functions.

When we take the limit of a function we are describing where they are going!

Let's see an example of this with the function f(x) = 3x^2 - 1

For this function I'm really curious what value the function is approaching as I use

x values close to the number 2.

Let's see this by using some inputs 1.9, 1.99, and 1.999.

When I use these, I get values such as 9.83, 10.8803, and 10.988003.

From these values is appears that the function is approaching 11.

So we say the limit of the function as x approaches 2, is 11.

Remember What I'm really saying here is that we can get arbitrarily close to the number

11, I just have to pick values that are sufficently close to 2 in order to do it.

Now at this point you might be thinking, that's fantastic, but couldn't you have found the

limit simplying by plugging 2 into the function.

Wouldn't that also give you 11?

In this instance the answer is yes, but the focus with a limit should be on what value

its approaching, and there are some functions where you simply can't plug in a number to

find the limit.

In otherwords, they are not always the same.

Let's cover this by going back to our pizza analogy.

Like before you have been struck with a craving for pizza so you are headed toward the fidge

for a quick snack.

Now in one scenario the fridge is there, loaded with pizza, and you can easily satify your

craving for pepperoni.

But in an alternate scenario the fridge is gone, possible stolen by pizza craving ninjas,

and you are left empty handed.

Even though both of these situations are completely different, your behavior leading up to them

is exactly the same.

In either case you were still approaching the fridge.

This is the key difference with limits, they are used to describe what value a function

is approaching.

They are not used to describe the value the function actually reaches.

Let's see how this works with yet another function.

Let's go ahead and use (x^2 - 4) / x-2 Like before we are interested in what value

the function approaches we use x values close to 2.

Let's go ahead and choose some inputs like 1.9, 1.99, and 1.999.

When we use these we get the values of 3.9, 3.99, and 3.999.

From these it appears that the function is approaching 4.

So again we say the limit of the function as x approaches 2, is 4.

If you try and find this value by instead plugging in 2, something strange happens.

When you plug 2 into the function, you get zero divided by zero.

This shows that the function doesn't actually ever get to 4.

In fact a quick look at the graph shows a hole right at 4.

Despite the hole, the behavior of the function leading up to it is the same.

Since the behavior is the same, we still say that the limit of the function as x approaches

2 is 4, even though it never actually gets there.

Hopefully both of these examples really highlight how limits focus on the behavior of a function,

what they get arbitrarily close to.

One thing we still have left to cover is what it exactly means when we say a function gets

"arbitrarily close" to a value.

But don't worry, We'll be able to tackle that tricky problem in the next video when we introduce

epsilon and delta.

Thanks for watching.

Hey, did you enjoy this video?

Don't forget to like it, and then subscribe to my channel!

If you want to know more about limits, you can watch a few examples here.

You can also move onto my next lecture video where I talk about the precise definition

of a limit using epsilon and delta!

For some of my other videos, don't forget to visit my web site: MySecretMathTutor.com

Thanks again for watching!