I agree that there is a problem with saying "find the limit using a table". You can certainly agree on a "most sensible guess" of the limit from a table of function values but you can't actually *find* a limit that way ... you could make another function that has those same values at each point of the table but has any value you want for the limit at the desired point (or, even, an undefined limit).
Finding a limit using a graph might be reasonable if you trust the makers of the graph to do something reasonable.
On another note, I recently came across the book "What Is Calculus About" by WW Sawyer (available from the MAA via the New Mathematical Library). I think it's an excellent book for building the intuition of what's really going on in calculus without getting overly bogged down in the details.