Even the stoutest hobbit felt all hope dissolve when he thought his master had died. How could he go on without Mr. Frodo? How could the wisest expect ole Samwise to go on?

And finish he must because he was all that was left of the Fellowship at that moment. 

He reluctantly said goodbye to his master, kissed him one last time.

Overhearing the orcs when they arrived to say that his master was not dead made him less doubtful of what had to be done.

Doubt melted away, hope returned. He had to finish the job with his master.