Q4

Find the first step in the following "Proof" that is incorrect, and provide an explanation why this is the case. As part of your explanation provide an example that shows the step is not valid.

Statement: If limxz f(x)g(x)=A then there are numbers B and C such that limxz f(x)=B and limxz g(x)=C and A=BC.

Proof: We are given that limxz f(x)g(x)=A. Then

limxz f(x)g(x)=(limxz f(x))(limxz g(x))(1)

Then let B be the value of limxz f(x) in (1) and let C be the value of limxzg(x) in (1) so that limxz f(x)=B and limxz g(x)=C. It needs to be shown that A=BC.