I had a question about finding limits of piecewise functions through graphs. I believe, I am missing something in my fundamentals about finding limits for these functions. Firstly, I would like to confirm that the empty dot represents a "hole" and the point is not included in the function of the line. For example, line closest to $h(x)$ is an example. Therefore, a coloured in dot represents a point that is on the graph e.g. $(3,2)$. So now for my main question
How do I determine the limit of a $x$ that has two $y$'s?
For example, the $\lim_{x\to 3^+} h(x)$ The back of the book says the answer is 3 but how can that be the case when, to my knowledge, an empty dot means that the point is hole? Or if that is not the case, then why is the answer not 2?
If someone could please clear this up for me, that would be great.
Thanks for the help!
P.S I just decided to add the questions in there for reference, I am not asking for anyone to solve them and please don't. I prefer that you give me the tools to do it myself or some hints.
2 Answers
$\begingroup$When evaluating limits, it doesn't matter whether the dot is filled in or not. Intuitively, the following two questions are the same:
What (if anything) is $\lim_{x\to 3^+}h(x)$?
If you were to approach the vertical line $x=3$ along the curve $y=h(x)$ from the right, what $y$-value (if any) would you approach?
Considered in the latter light, imagine tracing the path of the curve $y=h(x)$ and getting closer and closer to the line $x=3$. If we want to get to within a certain distance of the line $y=3$, then all we have to do is get close enough to the line $x=3$ from the right (this should be clear from the graph). That is what it means to say that $\lim_{x\to 3^+}h(x)=3$. We cannot do the same thing with the line $y=2$, since no matter how close to the line $x=3$ we get on the right, we keep tracing over points that are more than $1$ unit away from $y=2$.
$\endgroup$ 3 $\begingroup$When you say that $\lim_{x\to 3^+}h(x)=3$ it is not necessary to have $h(3)=3$.
$\endgroup$ 2
