P is true for sufficiently large x actually means:
- There exists a real number a such that P is true for all x ≥ a. This does not necessarily mean that any particular value for a is known, but only that such an a exists.
P is true for arbitrarily large x actually means:
- For every real number a, P is true for some values of x which are greater than a.
Food for thought
Since there are infinite prime numbers, we can say that prime numbers are arbitrarily large. However, not all sufficiently large numbers are prime.
In other words, for any number N you can find a prime number greater than N. However, there exists no number N such that all numbers > N are prime.