Is it true that if lim𝑥→6f(𝑥)=0 and lim𝑥→6g(𝑥)=0 then lim𝑥→6f(x)/g(x) does not exist?

Answer 1

No. An example is f(x) = (x-6)^2 g(x) = (x-6)

Then the ratio of f to g is x-6 which is zero in the limit.

It is true that f(x) / g(x) is undefined at x = 0. However the limit can exist.