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CStudent
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Webpage title: The Importance of Correct Order in the Definition of Limit
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In summary, the conversation discusses the conditions for a limit and how it is important to have the correct order in the definition. The condition $*\ \ \forall N\in\Bbb{N}\ \exists\epsilon>0 \text{ s.t. } \forall n>N \text{ we get } |a_n-L|<\epsilon$ does not imply that \lim_{n\to\infty}a_n = L, and the example of $a_n = 1$ and $L=0$ is given to illustrate this. It is then mentioned that the final condition, that if $b = |a_n-L| \to0\ \Longrightarrow\ \lim_{n\to\infty
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Opalg
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Hi CStudent, and welcome to MHB!
The condition $$*\ \ \forall N\in\Bbb{N}\ \exists\epsilon>0 \text{ s.t. } \forall n>N \text{ we get } |a_n-L|<\epsilon$$ does not imply that \(\displaystyle \lim_{n\to\infty}a_n = L.\)
For an example of how this can go wrong, suppose that $a_n = 1$ for all $n$ and that $L=0$. If you choose $\epsilon = 2$, the condition $|a_n-L|<\epsilon$ becomes $|1-0|<2$. That condition certainly holds for all $n$, but it is not true that \(\displaystyle \lim_{n\to\infty}a_n = L.\) In fact, \(\displaystyle \lim_{n\to\infty}a_n = 1\), which is different from $L$.
That example illustrates that in definitions like this it is very important to put things in the correct order. In the definition of limit, the choice of $\epsilon$ comes first, and then the choice of $N$ (which will usually depend on $\epsilon$) comes next. But in the condition $*$, you have put the $N$ before the $\epsilon$. That changes the whole meaning of the statement.
Your other condition $*$ is also not equivalent to the definition of limit.
But your final condition (the statement that \(\displaystyle b = |a_n-L| \to0\ \Longrightarrow\ \lim_{n\to\infty}a_n = L\)) is correct.
The condition $$*\ \ \forall N\in\Bbb{N}\ \exists\epsilon>0 \text{ s.t. } \forall n>N \text{ we get } |a_n-L|<\epsilon$$ does not imply that \(\displaystyle \lim_{n\to\infty}a_n = L.\)
For an example of how this can go wrong, suppose that $a_n = 1$ for all $n$ and that $L=0$. If you choose $\epsilon = 2$, the condition $|a_n-L|<\epsilon$ becomes $|1-0|<2$. That condition certainly holds for all $n$, but it is not true that \(\displaystyle \lim_{n\to\infty}a_n = L.\) In fact, \(\displaystyle \lim_{n\to\infty}a_n = 1\), which is different from $L$.
That example illustrates that in definitions like this it is very important to put things in the correct order. In the definition of limit, the choice of $\epsilon$ comes first, and then the choice of $N$ (which will usually depend on $\epsilon$) comes next. But in the condition $*$, you have put the $N$ before the $\epsilon$. That changes the whole meaning of the statement.
Your other condition $*$ is also not equivalent to the definition of limit.
But your final condition (the statement that \(\displaystyle b = |a_n-L| \to0\ \Longrightarrow\ \lim_{n\to\infty}a_n = L\)) is correct.
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CStudent
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Opalg said:
Great, thank you1
FAQ: Webpage title: The Importance of Correct Order in the Definition of Limit
What is the definition of the limit?
The limit is a fundamental concept in calculus that describes the behavior of a function as the input approaches a specific value. It is the value that a function "approaches" as the input gets closer and closer to a given value.
How is the limit of a function calculated?
The limit of a function is calculated by evaluating the function at values very close to the given input value. If the function approaches a single value, that value is the limit. If the function approaches different values from the left and right, or if it approaches infinity, then the limit does not exist.
What is the difference between a one-sided limit and a two-sided limit?
A one-sided limit only considers the behavior of a function as the input approaches a specific value from one direction (either left or right). A two-sided limit considers the behavior of a function as the input approaches a specific value from both directions, and the limit only exists if the function approaches the same value from both directions.
Can a function have a limit at a point where it is not defined?
Yes, a function can have a limit at a point where it is not defined. The limit only depends on the behavior of the function near the given point, not on the actual value of the function at that point.
How is the concept of the limit used in real-world applications?
The concept of the limit is used in a variety of real-world applications, such as calculating the instantaneous velocity of an object, determining the maximum capacity of a machine, and predicting the growth of a population. It is also used in engineering and physics to model and analyze the behavior of systems.