Question

Why in formal definition of limit function is defined in open interval?

Answer 1

a) For a∈Ra∈R and a function ff we write limx→af(x)=Llimx→af(x)=L provided limx→aSf(x)=Llimx→aSf(x)=Lfor some set S=J∖{a}S=J∖{a} where JJ is an open interval containing aa. limx→af(x)limx→af(x) is called the [two-sided] limit of ff at aa. Note ffneed not be defined at aa and, even if ff is defined at aa, the value f(a)f(a) need not equal limx→af(x)limx→af(x). In fact, f(a)=limx→af(x)f(a)=limx→af(x) if and only if ff is defined at aa and ff is continuous at aa.
(b) For a∈Ra∈R and a function ff we write limx→a+f(x)=Llimx→a+f(x)=L provided limx→aSf(x)=Llimx→aSf(x)=Lfor some open interval S=(a,b)S=(a,b).limx→a+f(x)limx→a+f(x) is the right-hand limit of ff at aa. Again ff need not be defined at a--_----------------------
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Answer 2

Answer:

$\LARGE{\underline{\underline{\bf{AnsweR \:  : -}}}}$

• The reason that open interval may be preferred is that the limit requires f to be defined on a topological neighbourhood of ∞. A neighbourhood of ∞ is a set that contains an open set containing ∞ and the basic open sets are open intervals