I've decided to teach myself calculus, I have a fairly solid knowledge of basic maths (did my A level last year) but I'm not sure if the book I'm using is particularly well written. I'm struggling to get my head around the example in pic related, any help would be appreciated
He's just giving you an example of how the limit can be visualized and rationalized. If you were to have a function y = x/sin(x), at x=0 the function is not defined (because we cannot divide by zero). Knowing this we have to use something called a limit. It just shows as a value approaches a point, the function in question converges to another point.
If you draw the graph you can see that it looks normal (it will even look like it passes through (0,1) ), but we still cannot define it for x=0. Graph the function and it might make more sense.
tl;dr - The picture is just giving an example of a limit, which you can think of as a fancy way of giving undefined points of a function legitimate values.