Everything about Asymptote totally explained
A
locally connected curve
A is said to be an
asymptote of the locally connected curve
B when the following is true:
» For any positive
, there exist
unbounded connected subsets (pieces of the respective curves)
and
, such that for every point in
its distance to the nearest point in
is lower than
.
In other words, as one moves along
B in some direction, the distance between it and the asymptote
A eventually becomes smaller than any distance that one may specify.
If a curve
A has the curve
B as an asymptote, one says that
A is
asymptotic to
B. Similarly
B is asymptotic to
A, so
A and
B are called
asymptotic.
Essentially, a linear asymptote is a straight line a line that a graph approaches, but doesn't become identical to.
Asymptotes, graphs and definitions
Meaning
Asymptotes are formally defined using
limits. There are many different cases that can be treated separately, such as linear asymptotes (below), although intuitively the two functions become arbitrarily close.
A specific example of linear asymptotes can be found in the
graph of the
function f(
x) = 1/
x, in which two asymptotes are seen: the horizontal line
y = 0 and the vertical line
x = 0.
There are multiple ways of interpreting asymptotic behavior. In particular the statement "
A function f
(x
) is said to be asymptotic to a function g
(x
) as x
→ ∞" has any of at least three distinct meanings:
- f(x) − g(x) → 0.
- f(x) / g(x) → 1.
- f(x) / g(x) has a nonzero limit.
More formally, curves
and
are asymptotic if and only if there exist continuous functions
Translations of known functions
If a known function has an asymptote (such as
y=0 for
f(x)=
ex), then the translations of it also have an asymptote.
If x=a is a vertical asymptote of f(x), then x=a+k is a vertical asymptote of f(x-h)+k
If y=b is a horizontal asymptote of f(x), then y=b+h is a horizontal asymptote of f(x-h)+k
For example, f(x)=ex-1+2 has horizontal asymptote y=0+2=2, and no vertical or oblique asymptotes.
Further Information
Get more info on 'Asymptote'.
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