so I have been working on limits and I thought I understood it, but my online homework says I have been incorrect. What am I doing wrong? Could someone explain limits to me?
I am asked to find the following and here are my answers:
$$\lim_{x\to 2-}= 3 \\
\lim_{x\to 2+}= 1 \\
\lim_{x\to 2}= \mathrm{Does~ not~ exist} \\
\lim_{x\to 0}= 3 \\
g(2)= 2 $$
Here is the graph:
Answering your questions from top to bottom:
The first one is asking for the left-hand limit (indicated by the minus sign). To find this you follow the graph of your function from the left of the curve to the right as x approaches 2. Doing this, you can clearly see you answer is correct.
The second asks for the right-hand limit (indicated by the plus sign) as x approaches 2. Following the same logic but from the other direction, we again find your answer to be correct.
The third is asking for the limit as x approaches 2. However, as we see in the above answers, the limit as x approaches 2 is different depending on the direction. Thus, we can see that there is no limit as x approaches 2. It is important to test the function from both sides of the limit.
Using the same logic as above, we can see that the left-hand limit of the function as x approaches 0 is equal to 3. However, we must also check to see if the right-hand limit is the same. Checking your graph, we can easily see the limit as x approaches 0 from the right is -1. As the limits differ depending on direction, the answer should be the same as the question above. The limit does not exist as x approaches 0.
Finally, this is asking for the value of the function at x = 2. Looking at your graph it easy to find the answer, which you have correctly said is 2.
Finding a limit generally means finding what value y is for a value of x. An easy method of finding a limit, if it exists, is the substitution method. All you have to do is substitute the x value that you want the limit for, into your function. Sometimes this may not be possible, as it may end up with the division of 0 for example. To fix this issue, you should sub values close to x, slowly getting closer and closer to x, and evaluate your function from both sides. For example, if you want the limit as x approaches 1 but evaluating x = 1 is impossible. First evaluate x = 0.9, then x = 0.99, then x = 0.999, etc. By doing this you will quickly see that the function approaches some value, which is your limit. Remember it is important to do this from both sides, so you must evaluate x = -0.9, x = -0.99, x = -0.999, to make sure the limit is the same as you approach x from both sides. Otherwise, an ordinary does not exist, as seen above.
On a semi-related note, just because the left-hand and right-hand limits are equal as they approach some value of x, it does not mean the function is continuous at this point.