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  2. List of limits - Wikipedia

    en.wikipedia.org/wiki/List_of_limits

    In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]

  3. Limit of a function - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_function

    In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point. A function is continuous at a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p. There is another type of limit of a function, namely the ...

  4. Indeterminate form - Wikipedia

    en.wikipedia.org/wiki/Indeterminate_form

    Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.

  5. Limit (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Limit_(mathematics)

    On one hand, the limit as n approaches infinity of a sequence {a n} is simply the limit at infinity of a function a(n) —defined on the natural numbers {n}. On the other hand, if X is the domain of a function f ( x ) and if the limit as n approaches infinity of f ( x n ) is L for every arbitrary sequence of points { x n } in X − x 0 which ...

  6. Uniform convergence - Wikipedia

    en.wikipedia.org/wiki/Uniform_convergence

    A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = ⁡ (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).

  7. Squeeze theorem - Wikipedia

    en.wikipedia.org/wiki/Squeeze_theorem

    The functions g and h are said to be lower and upper bounds (respectively) of f. Here, a is not required to lie in the interior of I. Indeed, if a is an endpoint of I, then the above limits are left- or right-hand limits. A similar statement holds for infinite intervals: for example, if I = (0, ∞), then the conclusion holds, taking the limits ...

  8. Thomae's function - Wikipedia

    en.wikipedia.org/wiki/Thomae's_function

    Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration. [ 4 ] Since every rational number has a unique representation with coprime (also termed relatively prime) p ∈ Z {\displaystyle p\in \mathbb {Z} } and q ∈ N {\displaystyle q\in \mathbb {N ...

  9. Extended real number line - Wikipedia

    en.wikipedia.org/wiki/Extended_real_number_line

    For example, the infinite sequence (,, …) of the natural numbers increases infinitively and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has + as its least upper bound and as its limit (an actual infinity).