A competitive price-taker firm would be willing to remain in the industry in the long run at zero economic profit because

It is covering accounting cost

Explanation:

In the long-run, there is no economic profit for a firm operating in a perfectly competitive market because there is free entry and exit.

In that same long run, the perfectly competitive firm is able to make an accounting profit. It is to be noted that economic profit is different from accounting profit. While economic profit is super-normal profit (excess profit), accounting profit is the normal profit.

When the super-normal profit is zero (in the long-run), the firm is still able to make its normal profit (accounting profit).

Hence, the business still remains profitable