Introduction

    Suppose f is a real valued function defined on a subset D of R. We are going to define limit of f (x) as x ∈ D approaches a point which is not necessarily in D.

    First we have to be clear about what we mean by the statement “x ∈ D approaches a point a”.

      Limit point of a set D ⊆ R

   Definition 1.2 Let D ⊆ R and a ∈ R.  Then a is said to be a limit point of D if for any δ > 0, the interval (a − δ, a + δ) contains at least one point from D other than possibly a, i.e.,

D ∩ {x ∈ R : 0 < |x − a| < δ} = ∅.

Lemma 1.2 The statements in the following can be easily verified:

• Every point in an interval is its limit
• If I is an open interval of finite length, then both the end points of I are limit points of
• The set of all limit points of an interval I of finite length consists of points from I together with its
• If D = {x ∈ R : 0 < |x| < 1}, then every point in the interval [−1, 1] is a limit point of
• If D = (0, 1) ∪ {2}, then 2 is not a limit point of The set of all limit points of D is the closed interval [0, 1]
•  

• If D = { 1 : n ∈ N}, then 0 is the only limit point of
• If D = {n/(n + 1) : n ∈ N}, then 1 is the only limit point of

For the later use, we introduce the following definition.