Need understanding on a sentence based on limits. I am studying calculus where there is a sentence in my book which is,

The definition of limit asserts that the entire graph of f above the interval $$N_2(p)$$ lies within this rectangle, except possibly for the point on the graph above p itself.

I can't find any meaning from this sentence as a whole. I need assistance.
Here is the graph of the function.

 A: The main difficulty in trying to answer your question is that I don't know whether your difficulties are with English, or with math, or with some combination of both. That said, here's an attempt to discuss the meaning of the diagram.
There’s probably more context before the sentence you quoted (I don’t have a copy of Apostol’s Calculus book to investigate this), so I’m going to guess what was probably said before this. You have a real-valued function $f(x)$ whose domain includes all points in an open interval containing $p,$ except possibly the point $p$ itself. Thus, while $f(p)$ might be undefined, we require that $f(x)$ is defined for all values of $x$ on both sides of $p$ that are sufficiently close to $p.$ For example, this requirement is satisfied if $f(x)$ exists for all values of $x$ such that $p - 0.0013 < x < p$ and $p < x < p + 0.0047.$
The assertion “the limit of $f(x)$ as $x \rightarrow p$ is equal to $A$” intuitively means that the graph of $y = f(x)$ comes together at the point $(p,A).$ That is, the portion of the graph lying on the left side of $x=p$ and the portion of the graph lying on the right side of $x=p$ join together at the point $(p,A).$ This can be rephrased in the following way. Imagine there is a vertical wall, infinitely high and infinitely low, that is erected on each side of the point $p.$ Now imagine moving both walls towards $p$ so that the distance between the walls gets smaller and smaller, without any positive lower limit (i.e. the walls get closer than $0.01$ apart, then closer than $0.001$ apart, then closer than $0.00073$ apart, etc.). (I'm reminded of the trash compactor scene in the first-made Star Wars movie.) The limit exists at $x=p$ means that the “vertical width” of the graph that lies between the walls also gets smaller and smaller, without any positive lower limit. That is, as you horizontally squeeze the graph together at $x=p,$ it must be the case that the graph is also vertically squeezed together. See this diagram for an example (limit doesn’t exist) in which the graph would NOT get vertically squeezed together in this manner.
In this discussion it is important to remember that the value of $f(p)$ is not considered, and in fact might not even exist. For example, for our purposes it does not matter if $(p,f(p))$ is very distant from $(p,A).$
Finally, to rephrase this even more precisely: Given any positive vertical distance $\epsilon,$ it is possible to choose a horizontal distance (this is a possible choice of $\delta$ in the $\epsilon$-$\delta$ definition of “limit”) between the walls so that the portion of the graph lying between the walls, except possibly the point $(p,f(p)),$ has a “vertical width” that is less than $\epsilon.$ In particular, the graph between the walls goes no higher than $A + \epsilon$ and the graph between the walls goes no lower than $A - \epsilon.$ (Keep in mind that the graph AT $x=p,$ namely the point $(p,f(p)),$ might be higher or lower than this.)
In the book’s diagram, the shaded interval on the $x$-axis indicates the values of $x$ corresponding to points on the graph that lie between the imagined vertical walls, and the shaded interval on the $y$-axis indicates the values of $y$ corresponding to points on the graph that lie between the imagined vertical walls. [Keep in mind that $p$ on the $x$-axis is not one of the $x$-values we have to worry about. That’s why in the diagram there is a circle in the graph at the point $(p,A)$ -- the circle shows where the graph tends to join together, while also indicating that we are not claiming that this point is actually on the graph.]
In the diagram, the limit of $f(x)$ as $x \rightarrow p$ is equal to $A$ means that no matter how vertically short we make the interval $N_1(A)$ on the $y$-axis (as long as $A$ lies within that open interval), we can find an interval $N_2(p)$ on the $x$-axis containing $p$ such that all the points on the graph corresponding to $x \in N_2(p)$ (except possibly $x=p)$ are such that they stay within the vertical bounds established by $N_1(A).$ In other words, all points on the graph corresponding to $x \in N_2(p)$ (except possibly $x=p)$ lie within the shaded rectangle shown in the diagram.
