In the Second Canto of the Śrīmad-Bhāgavatam (2.1.6) Śukadeva Gosvāmī says:

etāvān sāṅkhya-yogābhyāṁ svadharma-pariniṣṭhayā janma-lābhaḥ paraḥ puṁsām ante nārāyaṇa-smṛtiḥ

“The highest perfection of human life, achieved either by complete knowledge of matter and spirit, by acquirement of mystic powers, or by perfect discharge of one’s occupational duty, is to remember the Personality of Godhead at the end of life.” Somehow or other, Ajāmila consciously or unconsciously chanted the name of Nārāyaṇa at the time of death ( ante nārāyaṇa-smṛtiḥ ), and therefore he became all-perfect simply by concentrating his mind on the name of Nārāyaṇa.

It may also be concluded that Ajāmila, who was the son of a brāhmaṇa, was accustomed to worshiping Nārāyaṇa in his youth because in every brāhmaṇa’s house there is worship of the nārāyaṇa-śilā. This system is still present in India; in a rigid brāhmaṇa’s house, there is nārāyaṇa-sevā, worship of Nārāyaṇa. Therefore, although the contaminated Ajāmila was calling for his son, by concentrating his mind on the holy name of Nārāyaṇa he remembered the Nārāyaṇa he had very faithfully worshiped in his youth.

In this regard Śrīla Śrīdhara Svāmī expressed his verdict as follows: etac ca tad-upalālanādi-śrī-nārāyaṇa-namoccāraṇa-māhātmyena tad-bhaktir evābhūd iti siddhāntopayogitvenāpi draṣṭavyam. “According to the bhaktisiddhānta, it is to be analyzed that because Ajāmila constantly chanted his son’s name, Nārāyaṇa, he was elevated to the platform of bhakti, although he did not know it.” Similarly, Śrīla Vīrarāghava Ācārya gives this opinion: evaṁ vartamānaḥ sa dvijaḥ mṛtyu-kāle upasthite satyajṣo nārāyaṇākhye putra eva matiṁ cakāra matim āsaktām akarod ity arthaḥ. “Although at the time of death he was chanting the name of his son, he nevertheless concentrated his mind upon the holy name of Nārāyaṇa.” Śrīla Vijayadhvaja Tīrtha gives a similar opinion:

mṛtyu-kāle deha-viyoga-lakṣaṇa-kāle mṛtyoḥ sarva-doṣa-pāpa-harasya harer anugrahāt kāle datta-jṣāna-lakṣaṇe upasthite hṛdi prakāśite tanaye pūrṇa-jṣāne bāle paṣca-varṣa-kalpe prādeśa-mātre nārāyaṇāhvaye mūrti-viśeṣe matiṁ smaraṇa-samarthaṁ cittaṁ cakāra bhaktyāsmarad ity arthaḥ.

Directly or indirectly, Ajāmila factually remembered Nārāyaṇa at the time of death ( ante nārāyaṇa-smṛtiḥ ).