I'll be frank - Beethoven can be a little long winded in my opinion, however I have to acknowledge his refreshing chordal vocabulary, especially how he takes existing musical rules and twists them around. In particular, his Piano Sonata (C minor, Op. 13) has a great example of an augmented-sixth chord. In measure 46, on the first two beats Beethoven approaches the augmented-sixth chord with a big IV6. The key of the piece at this point is technically Eb major (relative major to C minor). This makes the IV6 spell out as Ab C Eb, C in the bass.
Immediately after the IV6 approach, Beethoven slams a French augmented-sixth chord into the half cadence. A typical French augmented-sixth chord in the key of Eb major is spelled Cb Eb F A natural, with Cb in the bass. This chord remains for beats 3 and 4 of measure 46 in a block chord. As necessary, le (Cb) resolves down in the next measure to Bb (sol) in contrary motion to the Fi (A natural) which resolves up to Bb also (sol). This contrary motion resolving Fi and le is the most important part of the augmented-sixth chord and it's role in the Half Cadence.
This whole instance in measure 46-47 is honestly, probably the most "text-book" example of an augmented-sixth chord in real life not only because of the spelling of the chords, but because of the block half-note rhythm of the section. Another really important part of these passages is the chromatic descent of the lowest note. In this example, La Le Sol (C Cb Bb) is a signal of this type of mode mixture.
A few bars later, in measures 49-50, Beethoven is extremely complex with his cadential material. As I mentioned earlier, regarding augmented-sixth chords, it is REALLY important that there is a fi resolving up to sol in contrary motion to le resolving down to sol. There is in fact a "quasi-augmented-sixth chord" from measure 49-50. A Db resolves down to a C while a B natural resolves up to a C. Since le is Db, the tonic note should be F.
In the case of most augmented-sixth chords, the chord that is resolved TO is a V chord. In measure 50, this should-be V chord is spelled Ab C Eb. This is unexpected, since with a typical augmented-sixth chord we would expect a V and in this case it should have been an F major chord. The strange thing is that this Ab chord can definitely be considered the dominant V chord of Db. Look at the bass note for example. It is a G which is technically a fi with regards to the key of Db.
There is a lot going on here. The cadential material in the treble clef resembles an augmented-sixth chord because of contrary motion between an apparent le and fi resolving to sol (C). This would in theory go to a C major chord as a V in the key of F major. However, the bass clef seems to be in a different key that shows a resemblance to a French augmented-sixth cadence in Db with a Re sustained over a fi-sol movement. Obviously this lacks a le-sol movement and is thus not a complete augmented-sixth chord either. Looking at what chord actually resolves of this is a mix of the two augmented-sixth chords that is pretty unexpected. It is an Ab major chord, which is IV. In my opinion, Beethoven took the basic structure of an augmented-sixth chord and altered it in order to resolve to a IV.
In conclusion, I believe that this is not a true augmented-sixth chord because in any way you look at it, you cant complete the figure. Undoubtedly this is an example of strange mode mixture, with a little hint of a secondary dominant that I can't really explain. In a performance, this part of the piano sonata most certainly demands close attention and adequate emphasis from the performer to really throw off the audience. As you will notice in the measures 50 and 51. The piece quickly flies through a rather unusual IV - V 7- I after this rather joggling mixture, which is quite humorous, though typical of Beethoven.