Definition:
Let A be a subset of R, 'a' be a limit point of A and f : A → R.
A real number 'I' is said to be a
limit of f at 'a', if to each given ε > 0,
∃ δ > 0 ∋ x ∈ A, 0 < | x – a | < δ
⇒ | f(x) – l | < ε.
It is denoted by f(x) = l
or f(x) → l as x → a.
If the function is defined on a deleted neighbourhood of a real number 'a',
then the limit of the function at 'a' can be defined as follows:
Let a, l ∈ R and f be a function defined on a
"deleted neighbourhood A of a".
Then l is said to be a limit of f at 'a' if
to each given ε > 0, ∃ δ > 0 ∋
x ∈ A, 0 < | x – a | < δ ⇒ | f(x) – l | < ε.
It is denoted by
f(x) = l or
f(x) → l as x → a.
A function f(x) is said to tend to a limit 'L' when 'x' tends to 'a', if the difference between f(x) and 'L' can be made as small as we please by making 'x' sufficiently near 'a'. We write:
f(x) = L or f(x) = L
The above is read as:
The limit of f(x) as 'x' approaches 'a' equals 'L'.
The notation means that the values of the function f(x) approach 'L' as the values of 'x' approach 'a' (but do not equal 'a').
Ex: Suppose f : (0, ∞) → R is defined by f(x) = √x.
Then f(x) = 0
Let ε > 0 be given. Choose δ = ε2. Then δ > 0.
For all x with x ∈ (0, ∞),
0 < | x | < δ
i.e., 0 < x < δ
we have | f(x) – 0 | = √x < √δ = ε
∴ √x = 0
Two simple observations:
(i) For a constant value 'c', (c) = c
(ii) For an identity function 'x', (x) = a