Well, your wording could use some work! For the first one, determining whether there is a limit at x= 0 and if so what it is, you say
"x approaching from the left is 1 and x approaching from the right is 1". Grammatically, that means that x becomes 1 which is not what you mean. What you mean is that "as x approaches 0 from the left the limit of the function is 1 and, as x approaches 0 from the right, the limit of the function is 1". Then you say "So I think x approaching 0 exists". Well, of course, x exists! But what you mean is "the limit as x approaches 0 exists". That is correct because the two one-sided limits exist and have the same value. The limit of the function, as x approaches 0, is 1.
(Actually the graph itself looks very strange. You have an open circle on the graph at (-1, 0) with a closed circle below it at (-1, -1). That's not a problem- it means that the value of f(-1) is NOT 0, even though the graph appears to pass through it- f(-1)= -1 instead. But you have a closed circle ON the graph at (0, 1) and an open circle at (0, 0). Why? Just drawing the graph through (0, 1) would be enough! That open circle at (0, 0) doesn't tell us anything. Perhaps that just done to "confuse" thing and see if you are being careful.)
For the second part, determining the limits at -2, -1, 0, 1, and 3, you say, first "c= -2 is 0; c= -1 is 0". What is 0?? You mean, of course, that if c= -2, then the limit there is 0 and, if c= -1, then the limit there is 0. You need to say that it is the limit you are giving. Both of those are correct by the way. With that proviso, that your answers could be word much more clearly, yes, these limits are correct.
For the third part, determining whether of not the function is continuous at -2, -1, 0, 1, and 3, you have the same poor wording: "c= -2 is continuous". Gramatically, that says that c is continuous which makes no sense! c is a number, not a function. What you mean of course is that "the function is continuous at c= -2". Get into the habit of writing exactly what you mean as complete sentences- if nothing else it will amaze and shock your teacher!
But in the last one, at 3, you have a terrible howler! You say "c= 3 is continuous" by which you mean, of course, that the function is continuous at 3. But you alread said, in part 2, that the limit of the function as not existing there. And the definition of "continuous" has three parts: A function, f, is said to be continuous at x= a if and only if
i) f(a) exists
ii) \(\displaystyle \lim_{x\to a} f(x)\) exists
iii) \(\displaystyle \lim_{x\to a} f(x)= f(x)\)
You have already said that (ii) is not true and it should be clear from the graph, which never reaches x= 3, that (i) is not true so that (iii) has no meaning.
sorry, had to repost part 3
