I will chase after lecturer's cap (5)
I believe the answer is:
Here is my best explanation:
'cap' is the definition.
(I've seen this before)
'i will chase after lecturer's' is the wordplay.
'will chase' becomes 'mit' (I can't justify this - if you can you should give a lot more credence to this answer).
'after' is a charade indicator (letters next to each other).
'lecturer' becomes 'l'.
'imit' after 'l' is 'LIMIT'.
(Other definitions for limit that I've seen before include "verge", "Bounds", "qualify?", "Restrict - term", "Furthest extent", "Utmost", "Restriction; boundary", "Maximum", "Constrain" and "Confine to specific boundary".)