I think that continuity should come first. If all functions were continuous, there would be no need for limits. Newton and Leibniz got along just fine without them. The modern definition of limit was originally part of the definition of continuity: limits solved the problem of continuity. Logically – that is, if you are proving things – limits must come first, since the definition of continuity depends on limits. But to see why limits are needed, you need to look at places where limits do not exist – functions that are not continuous. For some reason the new AP Precalculus course, while using limits often, seems to have made an effort not to mention continuity. So, it’s your choice which to teach first.
Why Limits? A brief note on why we need limits
Deltas & Epsilons – Why they are not required for AP Calculus
Finding Limits – A summary of ways to find a limit.
Dominance – A way of finding limits at infinity.
Unlimited – Does a limit equal to infinity define infinity?
Good Question 5: 1998 AB2/BC2 – Limit of infinity vs. DNE, and an alternative solution found by students.
Determining the Indeterminate – Indeterminate and Does Not Exist are not the same thing.
Determining the Indeterminate 2 – A different kind of indeterminate form.
Limit of Composite Functions – How to find limits of composite functions.
Infinite Musings – Limits equal to infinity really define infinity.
Which Came First? – Continuity or limits?
Continuity – The relationships between continuity and limits
Continuity 2 – a discussion of the definition
From One Side or the Other – on one-sided continuity and differentiability.
Right Answer, Wrong Question – thoughts on “continuous on its domain.”
Asymptotes – A note on horizontal asymptote, the manifestation of a limit at infinity.
Fun with Continuity – a fun but important function that is defined for all Real numbers, but continuous for none: the Dirichlet function
Continuous Fun – More on continuity and “continuous on its domain.”
The Intermediate Value Theorem (IVT) – a result of continuity
Intermediate Weather – An unexpected result: weather obeys the IVT!
Darboux’s Theorem – The Intermediate Value Theorem applies to derivatives (Not an AP Calculus topic, but interesting.)