Question

prove that if limit of function exists than it is unique​

Answer 1

Answer:

The limit of a function is unique if it exists. f(x) = L2 where L1,L2 ∈ R. For every ϵ > 0 there exist δ1,δ2 > 0 such that 0 < |x − c| < δ1 and x ∈ A implies that |f(x) − L1| < ϵ/2, 0 < |x − c| < δ2 and x ∈ A implies that |f(x) − L2| < ϵ/2. ... We can rephrase the ϵ-δ definition of limits in terms of neighborhoods.

Answer 2

Answer:

The limit of a function is unique if it exists.

LHL = f(a) = RHL                                                 [@x = a in f(x)]

f(x) = L2 where L1,L2 ∈ R. For every ϵ > 0 there exist δ1,δ2 > 0 such that 0 < |x − c| < δ1 and x ∈ A implies that |f(x) − L1| < ϵ/2, 0 < |x − c| < δ2 and x ∈ A implies that |f(x) − L2| < ϵ/2. ... We can rephrase the ϵ-δ definition of limits in terms of neighborhoods.