Functions with same limit at infinity | Limits and continuity | AP Calculus AB | Khan Academy

Khan Academy
29 Jun 201703:45
EducationalLearning
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TLDRThe video script discusses the concept of limits in calculus, emphasizing that an infinite number of functions can have the same limit as x approaches infinity. It illustrates this with examples of functions approaching the limit of three, showcasing both straightforward and more complex functions. The visual demonstration of graphs helps to convey how different functions can converge to the same value, highlighting the versatility and depth of mathematical analysis in understanding limits.

Takeaways
  • 🌟 The concept explained is about limits of functions as x approaches infinity, highlighting that an infinite number of functions can have the same limit.
  • 📈 The video provides examples of functions that all have a limit of three as x approaches infinity, demonstrating the general statement with specific cases.
  • 🔍 The script discusses the behavior of functions with very large x values, explaining that constant terms like +5 become insignificant compared to terms like x squared.
  • 📊 A graphical representation is used to illustrate how functions converge to their limits, with the example of a green graph approaching the horizontal asymptote y = 3.
  • 🚀 The video emphasizes the idea of convergence by showing that no matter how large x becomes, the function values get closer and closer to the limit, even if the rate of convergence varies.
  • 🤔 The script encourages viewers to ponder the nature of infinity and how it affects the behavior of functions, noting that even very large numbers like a trillion are insignificant in this context.
  • 🌐 The concept of oscillating functions around an asymptote is introduced, showing that functions can oscillate yet still approach a limit as x becomes infinitely large.
  • 📉 The video contrasts different functions with varying rates of convergence, using the example of a wild function involving natural logs that also approaches three as x approaches infinity.
  • 🔢 The importance of precision in mathematics is highlighted by discussing how functions can approach their limits to several decimal places, even if they do not quite reach the exact limit.
  • 📌 The video script serves as an educational tool to help viewers understand the flexibility and variety in the behavior of functions at infinity, reinforcing the principle that any function could have a limit, chosen arbitrarily in this case as three.
Q & A

  • What is the main concept discussed in the video?

    -The main concept discussed in the video is that there can be an infinite number of functions that have the same limit as x approaches infinity.

  • What does the instructor illustrate using the graph of a function?

    -The instructor illustrates that when x values are very large, the function approaches a certain value, in this case, three, and this can be true for an infinite number of functions.

  • How does the function 3x^2/x^2 behave as x approaches infinity?

    -As x approaches infinity, the function 3x^2/x^2 approaches the value 3, because the x^2 terms cancel out, leaving the constant 3.

  • What is the significance of the plus five in the function when x is very large?

    -When x is very large, the plus five in the function does not significantly affect the result, and the function gets closer to the value of 3x^2/x^2, which is 3.

  • How does the function involving natural logs behave as x approaches infinity?

    -The function involving natural logs also approaches the value of 3 as x approaches infinity, although it might do so at a slightly slower rate compared to other functions.

  • What did the instructor demonstrate with the oscillating function?

    -The instructor demonstrated that even functions that oscillate around the asymptote can approach a certain limit as x gets larger and larger, as long as the oscillations decrease in amplitude.

  • How does the rate of approach to the limit affect the overall limit value?

    -The rate of approach to the limit does not affect the overall limit value. Even if some functions approach the limit slower than others, the limit remains the same as long as the values get closer to the limit as x approaches infinity.

  • What does the instructor mean when they say 'the limit of the function as x approaches infinity'?

    -The instructor is referring to the value that the function gets closer and closer to as x increases without bound. This limit can be a real number, like 3 in the examples given, or it can be positive or negative infinity.

  • What is the importance of understanding the concept of limits in calculus?

    -Understanding the concept of limits is crucial in calculus because it forms the foundation for understanding continuity, derivatives, integrals, and many other advanced concepts. It helps in analyzing the behavior of functions and their properties at certain points or as input values change.

  • How does the video script help in visualizing the concept of limits?

    -The video script, combined with the visual representation of the function graphs, helps in visualizing how different functions can have the same limit as x approaches infinity. It shows that regardless of the initial behavior of the functions, they can all converge to the same value when x becomes very large.

Outlines
00:00
🌟 Understanding Limits at Infinity

The paragraph introduces the concept of limits at infinity, emphasizing that there can be an infinite number of functions that approach the same limit. The instructor uses the specific example of functions approaching a limit of three as x approaches infinity. It is highlighted that even with very large values of x, the actual function may differ but will tend towards the same limit. The visual aid of a graph is mentioned to illustrate how different functions converge towards the limit, with some functions approaching it faster than others. The idea is to demonstrate the variety of functions that can have similar asymptotic behavior at infinity.

Mindmap
Keywords
💡Limit
In the context of this video, 'limit' refers to the value that a function approaches as the input (x) becomes very large, specifically as x approaches infinity. The concept is crucial for understanding the behavior of functions at extreme values and is used to make predictions about their long-term trends. For instance, the video discusses functions that have the same limit of three as x approaches infinity, regardless of their initial behavior or complexity.
💡Function
A 'function' in mathematics is a relation that assigns a single output value to each input value. In this video, the term is used to describe various mathematical expressions that can be analyzed for their behavior as the input (x) approaches infinity. The main theme revolves around the idea that an infinite number of functions can have the same limit, highlighting the diversity of mathematical expressions that can lead to similar asymptotic behavior.
💡Asymptote
An 'asymptote' is a line that a function approaches but never actually reaches, becoming closer to it as the input (x) increases without bound. It is a useful concept for describing the long-term behavior of functions, especially when dealing with limits at infinity. In the video, the asymptote is the horizontal line y = 3, which all the discussed functions approach as x grows larger.
💡Infinity
In mathematics, 'infinity' is not a number but rather a concept that represents an unbounded quantity, something that can grow without limit. The video uses this concept to discuss the behavior of functions as their input (x) approaches infinity, focusing on how these functions tend towards a certain limit despite their potentially complex or different initial behaviors.
💡Natural Logs
Natural logs, or logarithms with base 'e' (where 'e' is a mathematical constant approximately equal to 2.71828), are a fundamental concept in calculus and mathematics. They are used to model growth and decay processes and are particularly relevant when dealing with exponential functions. In the video, a function involving natural logs is given as an example of a function that approaches the same limit as x approaches infinity, illustrating the diversity of functions that can exhibit similar asymptotic behavior.
💡Oscillating
In the context of the video, 'oscillating' refers to a function's behavior where it fluctuates above and below a certain value, in this case, the asymptote. This term is important for understanding how some functions may not approach their limit in a straightforward manner but instead move in a wave-like pattern around the limit before eventually getting closer to it as the input (x) becomes very large.
💡Trillion
A 'trillion' is a large numerical value, specifically 10^12 in the American system. In the video, this term is used to illustrate the concept of very large numbers and how they still pale in comparison to infinity. It helps to emphasize the idea that no matter how large a number is, infinity is always larger, and functions continue to approach their limits as x grows without bound.
💡Zoom
In the context of the video, 'zoom' refers to the action of adjusting the scale or perspective to view smaller details or larger portions of a graph. This is important for visualizing the behavior of functions as x approaches very large values and for understanding how closely the function approximates its limit or asymptote.
💡Decimal Places
The term 'decimal places' refers to the digits to the right of the decimal point in a number. It is used to express the precision or accuracy of a numerical value. In the video, the concept is used to discuss the precision with which the functions approach their limit, indicating that even though they may not have reached the exact limit, they can get extremely close to it, to several decimal places.
💡Graph
A 'graph' is a visual representation of the relationship between variables, typically used to illustrate the behavior of functions. In this video, graphs are essential for showing how different functions approach their limits as x becomes infinitely large. They provide a visual demonstration of the main theme of the video, which is the convergence of various functions to the same limit.
💡Behavior
In mathematics, 'behavior' refers to how a function or a system operates or performs under different conditions. In the video, the term is used to describe the way functions act as the input (x) approaches infinity, focusing on how they converge to a particular limit. Understanding the behavior of functions is crucial for analyzing their properties and making predictions about their long-term trends.
Highlights

The goal of the video is to understand that an infinite number of functions can have the same limit as x approaches infinity.

The general statement is that the limit of a function f(x) as x approaches infinity can be equal to a constant, in this case, three.

An example is given where a function gets closer to 3x^2/x^2, which equals three, as x becomes very large.

The plus five in the example function becomes insignificant as x increases, illustrating the concept of limits effectively.

The video visually demonstrates the concept by showing a graph where the function approaches the horizontal line y=3 as x approaches infinity.

Another function involving natural logarithms is introduced, showing that it also approaches three as x approaches infinity.

The rate of approach to the limit can vary among different functions, yet all tend towards the same limit in the infinite domain.

Functions can oscillate around the asymptote but still approach it as x increases without bound.

A zoomed-in view of the graph at x=14 demonstrates that all functions are approaching three, with some oscillating and others not.

Even at very large values like x=200, the graphs are still approaching the asymptote, though this requires significant zooming to observe.

The concept of limits is emphasized, showing that an infinite number of functions can have any constant limit as x approaches infinity.

The video serves as a visual tool to understand the abstract concept of limits in calculus.

The arbitrary choice of three as the limit illustrates that any constant could be the limit of a function as x approaches infinity.

The video provides multiple examples to reinforce the idea of limits and their applicability to a wide range of functions.

The importance of understanding the behavior of functions as x approaches very large numbers is highlighted.

The video is part of a series that will explore the reasons behind the behavior of functions at infinity in more depth.

The visual representation helps to clarify the abstract mathematical concept, making it more accessible to learners.

Transcripts

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Calculus ConceptsLimits InfinitudeFunction ConvergenceMathematical ExamplesGraph AnalysisAsymptotic BehaviorEducational ContentInfinite SeriesTrigonometric FunctionsLogarithmic Functions