Question

exercise 5.2.6. let g be defined on an interval a, and let c ∈ a. (a) explain why g′(c) in definition 5.2.1 could have been given by g′(c)

Answer 1

From definition 5.2.1:

g'(c) = limₓ→c  g(x) ₋ g(c)/x₋c

Let h = x ₋ c

Then x = c₊h

substituting in the above defination we have:

g'(c) = limc₊h→c  g(c₊h) ₋ g(c)/c₊h₋c

=  limc₊h→c g(c₊h) ₋ g(c)/h

x → c ⇒ x ₋ c → 0 ⇒ h → 0

Hence we have,

limc₊h→c  g(c₊h) ₋ g(c)/h

Therefore we have defined clearly why g′(c) in definition 5.2.1 could have been given by g′(c).

Learn more about limits and continuity here:

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