Use the definition of continuity to determine whether f is continuous at a. f(x) = 5x^4 - 9x^3 + x - 7a = 7

will be continuous at if
(i) exists,
(ii) exists, and
(iii) .

The second condition is immediate, since has a finite value. The other two conditions can be established by proving that the limit of the function as is indeed the value of . That is, we must prove that for any , we can find such that



Now,



Notice that when , we have . By the polynomial remainder theorem, we know that is then a factor of this polynomial. Indeed, we can write



This is the quantity that we do not want exceeding . Suppose we focus our attention on small values . For instance, say we restrict to be no larger than 1, i.e. . Under this condition, we have



Now, by the triangle inequality,



If , then this quantity is moreover bounded such that



To recap, fixing would force , which makes



and we want this quantity to be smaller than , so



which suggests that we could set . But if is given such that the above inequality fails for , then we can always fall back on , for which we know the inequality will hold. Therefore, we should ultimately choose the smaller of the two, i.e. set .

You would just need to formalize this proof to complete it, but you have all the groundwork laid out above. At any rate, you would end up proving the limit above, and ultimately establish that is indeed continuous at .

The first one is correct