Division is an operation carried out on two numbers, infinity isn't a number, therefore dividing infinity is meaningless.
Assuming it was possible to have a hole to the centre of the earth and a rope that did not melt the answer to the question would depend upon the strength of the rope because as the rope was lowered into the hole the weight of the rope elements would decrease with depth, as gravity decreases with depth, and the breaking strain of the rope may never be reached.
there's only one thing in the universe that could be described as limitless and that's yer imagination. apply that and you find the answer you seek.