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Can someone rephrase the title question into something more meaningful in terms of Calculus/Analysis?
Just to clarify, here we would say that x is the limit, not that x "has" a limit!Fredrik said:Suppose that ##x_1,x_2,\dots## is a sequence of points. A point ##x## is said to be a limit of that sequence if every open neighborhood of ##x## (i.e. every open set that contains ##x##) contains all but a finite number of the points in the sequence.
Fredrik said:Suppose that ##x_1,x_2,\dots## is a sequence of points. A point ##x## is said to be a limit of that sequence if every open neighborhood of ##x## (i.e. every open set that contains ##x##) contains all but a finite number of the points in the sequence.
pwsnafu said:There is a difference between the individual point ##x## and the sequence ##(x,x,x,\ldots)##. The former the does not have a limit but the latter does (and the limit is ##x##).
A point limit in calculus refers to the value that a function approaches as a specific variable approaches a given point. It is an important concept in calculus and is used to determine the behavior of a function near a specific point.
Calculus provides a set of tools and techniques to analyze the behavior of a function at a specific point. By using the concepts of continuity, derivatives, and limits, calculus can help determine the value of a point limit and understand the behavior of a function near that point.
No, calculus cannot determine point limits for all functions. There are some functions that are not continuous or differentiable at a given point, making it impossible to calculate the point limit using calculus. In these cases, other mathematical methods may be used to determine the point limit.
In calculus, infinity is often used as a way to describe the behavior of a function at a specific point. A point limit can approach infinity or negative infinity, indicating that the function is increasing or decreasing without bound near that point. Calculus helps to determine these infinite limits and understand the behavior of the function at those points.
Yes, calculus is commonly used in real-world applications to determine point limits. For example, it can be used in physics to calculate the velocity or acceleration of an object at a specific point in time, or in economics to determine the maximum profit or loss of a business at a certain point in production. Calculus is a powerful tool for understanding and analyzing the behavior of functions in real-world scenarios.