The finding of Sita.

Hanuman with his gang went searching for Sita and he couldn't find her. In this point in time, the vanaras entered a cave and then found a yogini performing tapas there. The vanaras stood there with clasped hands waiting her to finish her penance. Once she did, they asked her if she knew about the whereabouts of Sita. She told them that they vulture Sampathi in an adjacent cave might have knowledge of the whereabouts of Sita. Then they all walked upto Sampathi.
In this context, the contribution of Jatayu's brother, Sampaati, is worth mentioning. Jatayu and Sampaati, when young, used to compete as to who could fly higher. On one such instance Jatayu flew so high that he was about to get seared by sun's flames. Sampaati saved his brother by spreading his own wings and thus shielding Jatayu from the hot flames. In the process, Sampaati himself got injured and lost his wings. As a result Sampaati lived wingless for the rest of his life.
Both Jatayu and Sampathi were sons of Aruna and nephews of Jatayu.
According to the solar mythology of Hinduism, the legend states that Vinata was one of the wives of rishi Kashyapa, and she bore him two sons, named Aruṇá and Garuda. Kashyapa was promised that her sons would be powerful if she waited for them to awake. However, her impatience to hatch them took root, and she broke one of them. From the broken egg a flash of lighting, Aruṇá, emerged sprang forth. He was as radiant and reddish as the morning sun. But, due to the premature breaking of the egg, Aruṇá was not as bright as the noon sun as he was promised to be. Aruṇá's brother, Garuda, was born regularly, and eventually became the main vehicle of Vishnu.
Aruṇá is sometimes considered a part of Surya, as he is the vision and driving force behind its path through the sky. In some stories, Aruṇá drives the chariot of Surya, while in others, he is a manifestation of Surya, serving as a sign of the coming of the Sun God.
This Sampathi met the vanaras who came upto him and asked him if he knew the whereabouts of Sita. Then he told them that he saw a woman crying for help when Ravana was kidnapping her on his Pushpaka. Sadly, due to his inability to fly, he couldn't go for help. He guessed that must have been Sita. On asking for where Ravana lived, they were told he lived in Lanka, 100 yojanas away from Bharatavarsha.
A yojana is a Vedic measure of distance used in ancient India. The exact measurement is disputed amongst scholars with distances being given between 6 to 15 km (4 and 9 miles).
Now, someone had to leap through the distance.
Now, none of the vanaras are confident to jump that far. Angada says 80 yojanas, Neel says 70, Nala says 60.
Jambavantha (also known as Jambavan or Jamvanta) is a bear in Hinduism and believe to lived from Krita Yuga to Dvapara Yuga. He is also considered King of bears and first son of Brahma, before humans were created. Jambavan was present when the churning of the ocean took place and was supposed to have circled the world seven times. Jambavan says that on his youth days he would have jumped 200, but that now he couldn't manage more than 90.
Now hopes turn to Hanuman.
Hanuman was born in Kapisthal (Currently known as Kaithal, a place in Haryana) in the Treta Yuga, to 'Anjana', a female vanara. Anjana was actually an apsara or a celestial being, named 'Punjikasthala', who, due to a curse, was born on the earth as a female vanara. The curse was to be removed upon her giving birth to an incarnation of Lord Shiva.
Along with Kesari, Anjana performed intense prayers to Shiva to beget Him as her Child. Pleased with their devotion, Shiva granted them the boon they sought.
Different stories are told as to Hanuman's birth. One is that at the time that Anjana was worshipping Lord Shiva, elsewhere, Dashrath, the king of Ayodhya, was performing the Putrakama Yagna in order to have children. As a result, he received some sacred pudding, to be shared by his three wives, leading to the births of Lord Rama, Lakshmana, Bharata and Shatrughna. By divine ordinance, a kite snatched a fragment of that pudding, and dropped it while flying over the forest where Anjana was engaged in worship. Vayu, the Hindu deity of the wind, delivered the falling pudding to the outstretched hands of Anjana, who consumed it. Hanuman was born to her as a result.
Being Anjana's son, Hanuman is also called Anjaneya (pronounced Aanjanèya), which literally means "arising from Anjani".
Hanuman is also considered as the incaranation of Shiva or reflection of Shiva. When Ravana tried to enter the Himalayas (the abode of Shiva) Nandi stopped him and Ravana called Nandi a monkey. Nandi in return cursed Ravana: monkeys would help destroy him. In the process word monkey along with panchaksram "Om Namashivaya" hit Shiva. Shiva, to give respect to his devotee, took the form of a vanara, Hanuman.
As a child, assuming the sun to be a ripe mango, he once took flight to catch hold of it to eat. Indra, the king of devas observed this. He hurled his weapon, the Vajra (thunderbolt) at Hanuman, which struck his jaw. He fell back down to the earth and became unconscious. Upset, Vayu went into seclusion, taking the atmosphere with him. As living beings began to get asphyxiated, to pacify Vayu, Indra withdrew the effect of his thunderbolt, and the devas revived Hanuman and blessed him with multiple boons. However, a permanent mark was left on his chin (hanuhH in Sanskrit).
On ascertaining Surya, the Hindu deity of the sun to be an all-knowing teacher, Hanuman raised his body into an orbit around the sun and requested that Surya accept him as a student. But Surya refused, claiming that as he always had to be on the move in his chariot, it would be impossible for Hanuman to learn effectively. Undeterred by Surya's refusal, Hanuman enlarged his body; he placed one leg on the eastern ranges and the other on the western ranges and with his face turned toward the sun, made his request again. Pleased by his persistence, Surya accepted. Hanuman then moved (backwards, to remain facing Surya) continuously with his teacher, and learned all of the latter's knowledge. When Hanuman then requested Surya to quote his "guru-dakshina" (teacher's fee), the latter refused, saying that the pleasure of teaching one as dedicated as him was the fee in itself. But Hanuman insisted, and it was then that Surya asked him to help his (Surya's) spiritual son Sugriva. Hanuman's choice of Surya as his teacher is said to signify Surya as a Karma Saakshi, an eternal witness of all deeds.
Hanuman was mischievous in his childhood, and sometimes teased the meditating sages in the forests by snatching their personal belongings and by disturbing their well-arranged articles of worship. Finding his antics unbearable, but realising that Hanuman was but a child, (albeit invincible), the sages placed a mild curse on him. By this curse Hanuman forgot his own prowess, and recollected it only when others reminded him about it.
Then, Jambavan reminds him of his unearthly prowess. Hanuman then recollects his own powers, enlarges his body and flies across the ocean. On his way, he encounters a mountain Mandara that rises from the sea, proclaims that it owed his father a debt and asks him to rest a while before proceeding. Not wanting to waste any time, Hanuman thanks the mountain and carries on. He then encounters a sea-monster who challenges him to enter her mouth. When Hanuman outwits her mouth and coming out of her ears, she admits that it was merely a test of his courage. Finally on killing Simhika, a shadow-eater rakshasa, he reached Lanka.
On reaching Lanka, he reduces his size to an infinitesimally small size, and then looks at the beautiful city. Then he pities saying the beautiful city has to be destroyed due to Ravana's arrogance. Then, he moves into Ravana's palace. He sees the beautiful lamps glowing in the night, the beautiful chambeliers, the huge Kumbhakarna snoring, the noble Vibishana spending his nocturnal hours on the floor, and then the cruel Ravana, sleeping with a beautiful woman in his chambers. He first doubts if it is his Sita devi who has yielded to his attempts to make her his mistress, but then rubbishes such thoughts, and decides that it is Mandodari, Ravana's chief queen. Mandodari was the beautiful and pious daughter of the King of Danavas, Mayasura and the celestial dancer, Apsara Hema. She was the mother of Indrajit, who earned the name after defeating Indra, the King of the Devas.
Hanuman saw the powerful Indrajit sleeping soundly. He feels a pang of sorrow thinking about the forthcoming destruction of such warriors.
Then slowly he crawls to the Ashoka Vana. There he sees a beautiful lady in saffron, hair unkept, crying as Ravana forces her to marry him. Hanuman makes up his mind that she is Sita. Ravana tells her that if she doesn't give her approval to their alliance in another three months, she would be killed and her flesh would serve as appetizers for the flesh hungry Lankan demons.
Then, he leaves. Immediately Hanuman jumps down and then introduces himself to Sita, but she refuses to believe him and thinks of him to be an ally of Ravana. Then, he shows her a ring which Rama gave her as proof and seeing which she believes him and gives him her choodamani which she asks him to give Rama. He asks the devi to come with her. But then she rejects it saying, she has been dishonoured by being held captive and she will come back only if her husband comes back, vanquishes Ravana and takes her back.
Then Hanuman takes her blessing and starts his vanquish on the Ashoka Vana. Then, the Lankan troops come to stop him but in vain. Akshay Kumara, a son of Ravana, comes to deal with Hanuman but is killed by the Vanara god. Then Indrajith comes to fight. Hanuman realises that he wouldn't get a better chance to meet the ten-headed Ravana and so gets voluntarily bound by Indrajith's arrows. Then, the demon troops drag Hanuman to Ravana's court where he is sitting in full splendour, the ten-headed wonder. He asks why Hanuman damaged the beautiful Ashoka Vana. He tells him the reason of his visit and also fortells the vanquishing of Lanka in the hands of Rama, if he did not yield to allowing Sita to go home to Rama. An enraged Ravana orders to burn a monkey's most valued part of the body, his tail. Vibhishana requests Ravana not to do this, telling him that a messenger shouldn't be treated in that manner. Ravana does not heed to the advice and orders for the tail to be covered with cloth and then it being set to fire. Instantaneously Hanuman enlarges in size, and the covering of the tail with cloth becomes a mammoth task. But still they manage to do it and set fire to it. As soon as this is done, Anjaneya jumps out, goes across Lanka setting all places in Lanka to fire, leaving Ashoka Vana alone as Sita devi is there, takes leave of her and reaches the ocean where he plunges his burning tail into the sea. With this relief, he jumps and reaches the shore where the Vanaras are waiting. Then, they go to Rama, and Anjaneya says, "Saw I Sita!!" He says the saw first because he doesn't want to give Rama even a moment's anxiety regarding the matter of him having met Sita or not. He gives Rama, Sita's choodamani, which on seeing, Rama breaks into tears, in remembrance of his beloved. Lakshmana consoles him, and with this, Rama gets the urge to break the next frontier called Lanka.