Learn English – Where does “patching through” come from

"To bite the dust" means to die or to fail (see e.g. Wiktionary). Picture someone falling down, wounded or dead, quite literally biting the dust (soil, ground, earth). Etymonline says that the first recorded appearance of the phrase is from 1750. The Phrase Finder supplies it as follows:

The earliest citation of the 'bite the dust' version [of the earlier phrase 'lick the dust', from the Bible] is from 1750 by the Scottish author Tobias Smollett , in his Adventures of Gil Blas of Santillane:

"We made two of them bite the dust, and the others betake themselves to flight."

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[Samuel Butler's 19th century translation of Homer's The Iliad] contains a reference to 'bite the dust' in these lines:

"Grant that my sword may pierce the shirt of Hector about his heart, and that full many of his comrades may bite the dust as they fall dying round him."

Whether that can be counted as an 8th century BC origin for 'bite the dust' is open to question and some would say that it was Butler's use of the phrase rather than Homer's.

This one is actually quite straightforward. It alludes to going out on a branch of a tree. Etymonline says that the figurative sense is from 1897. The Phrase Finder supplies a quote from 1895:

The first uses of it in a figurative sense, with no reference to actual trees or climbing, come from the USA at the end of the 19th century. For example, the Steubenville Daily Herald, October 1895:

[...] If we get the 14 votes of Hamilton we've got 'em out on a limb. All we've got to do then is shake it or saw it off.