Infinito . If for any number N, as large as you want, that there exists such δ (N), for 0 <Ι x – a Ι <δ (N) the inequality Ιf (x) Ι> N is verified, the function f (x ) is called infinite (infinitely large) when x → a. Similarly, f (x) is determined as infinite (infinitely large) when x → ∞.
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- 1 Concept
- 2 Fundamental infinities
- 3 Comparison of infinities
- 1 Comparison of fundamental infinities
- 4 See also
- 5 Sources
This concept of infinity is used in Mathematical Analysis when we want to express that the terms of an ordered sequence , or the values that a function takes when the dependent variable takes values close to one previously set, “diverges” (“tends to infinity”, or its limit is infinite). In this context, considera is considered to represent the limit that tends to infinity and 0 to the limit when it tends to 0; and not to number 0).
Example: lim x-> 2 3 / (x-2) = ∞ => 3 / (x-2) is infinity when x tends to 2, (x → 2).
- Logarithmic infinity:
- Potential infinity, natural n and n ≠ 0
- Exponential infinity, a belonging to R +
- Exponential potential infinity, natural n and n ≠ 0
When comparing infinities all functions tend to infinity, then we have the following cases, the highest order is for the function that grows the fastest to infinity. Let the functions f (x) and g (x) be two infinites on a.
- F (x) and g (x) are said to have the same order if
- The order of f (x) is said to be greater than the order of g (x) if
- The order of f (x) is said to be less than the order of g (x) if
- When there is no limit, it is said that infinities are not comparable.
Comparison of fundamental infinities
The following are shown according to orders of the type of infinity, from highest to lowest:
- order of type x kx> order of type b x > order of type x m > order of type log a x c
Example of infinity orders
- Orders of type b x: 4 x and 1.5 x is greater than the one with the largest base 4 x > 1.5 x
- Orders of the type x m: 2x 3 , x 2 and x 1/2 is greater the one with the highest degree 2x 3 > x 2 > x 1/2
Sorting the above types in descending order: 4 x > 1,5 x > 2x 3 > x 2 > x 1/2 > log 2 x
The following example shows how a limit is calculated applying the concept of infinity orders.
- order x m> order log a x c , then x> lnx which corresponds to the case in which the order of f (x) is less than the order of g (x) so that it grows faster at infinity resulting in:
- Equivalent infinities: Two infinities f (x) and g (x) are said to be equivalent if the
- The sum of two infinites of different order is equivalent to the infinity of greater order.