Continuous functions/Limits and continuity
Uhm, yeah, so I was studying for my test and realized that there's a lot I really didn't understand. This is one of those things:
Instructions:
Find a value for a so that the function is continuous.
1. f(x) = {2x + 3, x ≤ 2; ax + 1, x > 2}
2. f(x) = {4 - x^2, x < -1; ax^2 - 1, x >= 1}
3. f(x) = {x^2 + x + a, x < 1; x^3, x >= 1}
(This thing here: >= is a greater than or equals to sign since I couldn't figure out how to make it and I couldn't find anything to copy-paste off of the internet.)
The other problem I had was a little different.
Instructions:
a) Draw the graph of f.
b) At what points c in the domain of f does the limit exist?
c) At what points c does only the left-hand limit exist?
d) At what points c does only the right-hand limit exist?
Problem:
4. f(x) = {x, -1 ≤ x < 0, or 0 < x ≤ 1; 1, x=0; 0, x<-1, or x >1}
My problem here is drawing the graph. It's the 'or's' that throw me off. I'm fairly certain that if I were to figure out the graph, that I could easily do the rest of the problem. I may be wrong, of course...
(Also, if it's not too much trouble, and I'm sorry if it sounds a bit pretentious especially since I'm asking for help, but if you could maybe do part of it or explain how it goes and allow me to figure it out for myself? It is for a test, as I said, and I want to make sure that I perfectly understand it. But y'know, only if it's not too much trouble.)
Also, I was wondering another thing, though this is just a general question or comment. Does it break the rules that I'm putting so many problems/questions in one post or is that still okay? I'm unsure.
