How could Shri Rama discharge arrows continuously in the war with Ravana?

 

Shri Rama and Lakshmana were carrying 2 quivers, from which inexhaustible arrows will emerge.

There are 2 versions available in the Ramayana, as to how Shri Rama got 2 quivers.

a) In Ayodhya Kanda, after Lakshmana insists to accompany Shri Rama and Sita to the Forest, Shri Rama told him to fetch 2 bows, 2 quivers with inexhaustible arrows, 2 swords given by Janaka to them and were kept in Sage Vasistha’s ashrama.

ये च राज्ञो ददौ दिव्ये महात्मा वरुणः स्वयम् |
जनकस्य महा यज्ञे धनुषी रौद्र दर्शने || २-३१-२७
अभेद्य कवचे दिव्ये तूणी च अक्षय सायकौ |
आदित्य विमलौ च उभौ खड्गौ हेम परिष्कृतौ || २-३१-२८
सत्कृत्य निहितम् सर्वम् एतत् आचार्य सद्मनि |
स त्वम् आयुधम् आदाय क्षिप्रम् आव्रज लक्ष्मण || २-३१-२९

"Oh, Lakshma! At a grand sacrifice perfomed by Janaka, the great-souled Varuna(god of water) personally gave heavenly bows which are dreadful to look at, divine impenetrable pices of armour, quivers containing an inexhanstible stock of arms, two swords decked with gold and with spotless lustre like that of a sun - all these were kept at the residence of our receptor Vasista, after paying due reverence. Take all those arms and return soon."

b) In Aranya Kanda, Sage Agastya gives One DIVINE bow and 2 quivers, with inexhaustible arrows, to Shri Rama.

इदम् दिव्यम् महत् चापम् हेम वज्र विभूषितम् |
वैष्णवम् पुरुषव्याघ्र निर्मितम् विश्वकर्मणा || ३-१२-३२
अमोघः सूर्य संकाशो ब्रह्मदत्तः शर उत्तमः |

"This sacred bow that is decorated with gold and diamonds is crafted by the divine architect Vishvakarma pertains to Vishnu, and these unwasteful arrows equalling sun in their blaze are the gift of Brahma.

दत्तो मम महेन्द्रेण तूणी च अक्षय सायकौ || ३-१२-३३
संपूर्णौ निशितैः बाणैः ज्वलद्भिः इव पावकैः |
महा रजत कोशो अयम् असिः हेमविभूषितः || ३-१२-३४

"Also these two inexhaustible quivers packed with arrows that have the blaze of Ritual-fire, and this sword decorated in gold together with its sheath made up of excellent golden are once given to me by Indra.

एवम् उक्त्वा महा तेजाः समस्तम् तत् वर आयुधम् |
दत्त्वा रामाय भगवान् अगस्त्यः पुनर् अब्रवीत् || ३-१२-३७

On saying thus that great resplendent and godly sage Agastya, gave all of those best weapons to Rama, and again spoke to Rama

——

I personally prefer the 1st version, ie., 2 bows, 2 quivers with inexhaustible arrows, 2 swords given by Janaka to them, as 2nd version is an interpolated one.