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If x is a cluster point for the sequence {x_n}, there is a subsequence of {x_n} whose limit is x. The subsequence can be constructed by choosing a sequence of shrinking neighborhoods V_k about x. Since x is a cluster point, each such neighborhood contains infinitely many elements of {x_n}. By choosing a new point x_k from each neighborhood V_k, we get a subsequence {x_k} of {x_n} with lim x_k = x.
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Is there a limit to the number of lines that can intersect at one point?
No limit.
How many planes can intersect a line at a given point?
an infinite number; no limit
How many lines can end in the same point?
I assume you mean that the line goes through a certain point. There is no limit to how many lines you can have through the same point.
When does the derivative of a function exist at a given point?
Let f be a function and a be the given point you are considering. Then,f(x) - f(a)---------------(x-a)is the difference quotient. If the limit as x approaches a exists, then the function is differentiable at a, or we say the derivative exists at a. If that limit does not exist, then the derivative does not exist at that point.
What is used to describe a set of data where the measures cluster or concentrate at a point?
The answer depends on the type of distribution for the data. It could be the modal class.
