Arbitrarily large and sufficiently large are both mathematical concepts.

Definitions

P is true for sufficiently large x actually means:

There exists a real number a such that P is true for all xa. 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.

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"Arbitrarily Large vs Sufficiently Large." Diffen.com. Diffen LLC, n.d. Web. 15 Nov 2017. < >