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As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as close to 2 as one could wish—by making x sufficiently large. So in this case, the limit of f ( x ) as x approaches infinity is 2 , or in mathematical notation, lim x → ∞ 2 x − 1 x = 2. {\displaystyle \lim _{x\to \infty }{\frac {2x-1}{x}}=2.}
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
It was published in 1821 by Cauchy, [1] but remained relatively unknown until Hadamard rediscovered it. [2] Hadamard's first publication of this result was in 1888; [ 3 ] he also included it as part of his 1892 Ph.D. thesis.
A non-degenerate random variable Z is α-stable for some 0 < α ≤ 2 if and only if there is an independent, identically distributed sequence of random variables X 1, X 2, X 3, ... and constants a n > 0, b n ∈ ℝ with a n (X 1 + ... + X n) − b n → Z.