hehehe - I am not the guru in the group!
Well, perhaps I am blending the film and the book this morning... On a quick skim through... I didn't find it... I found one passage that on its own makes it look like Frodo's presumption:
Frodo rose to his feet. A great weariness was on him, but his will was firm and his heart lighter. He spoke aloud to himself. `I will do now what I must,' he said. 'This at least is plain: the evil of the Ring is already at work even in the Company, and the Ring must leave them before it does more harm. I will go alone. Some I cannot trust, and those I can trust are too dear to me: poor old Sam, and Merry and Pippin. Strider, too: his heart yearns for Minas Tirith, and he will be needed there, now Boromir has fallen into evil. I will go alone. At once.'