0 Divided By Infinity Math
Since g x approaches infinity as x approaches a as x gets close to a g x 1.
0 divided by infinity math. Thus as x gets close to a 0 f x g x f x. Thus the answer to your question is 0. Regardless of what large number we re dividing by our answer is 0 and by letting this large number increase as much as we please tending to infinity the answer is still 0. Also what is not defined is zero divided by zero.
The infinity signs change when dividing by 0 instead. You might also look at my answer to a questiion about 1 infinity. There are two zeroes. In ieee 754 arithmetic a 0 is positive infinity when a is positive negative infinity when a is negative and nan when a 0.
If this is what you mean by dividing zero by infinity then it is not indeterminate it is zero. Thus f x g x must also approach zero as x approaches a. Any expression divided by infinity is equal to zero. 0 positive zero and 0 negative zero and this removes any ambiguity when dividing.
If you take 0 and divide by 10000 what would you get.