I need help with chapter 5 of Spivak calculus I am currently going through chapter 5 of Spivak's book on calculus. That chapter is on limits. I have done the entire chapter, however, I feel I have not completely grasped the concept of limits. In particular, my problem is in proving that the limit of a function does not exist. Moreover, another problem I am facing is understanding the concept of the very definition of limits. That is: if $f$ tends towards the limit $l$ close to \alpha, then for every $\epsilon>0$ there is a $\delta>0$  such that, for every x, if $0<|x - \alpha|<δ$, then $0<|f(x) - l|<\epsilon$ .Can someone please suggest some further reading I should do?
Awaiting for your reply
Andrew
 A: You can consult other books then return to it later. 
But I'll give you some ideas on limits that I hope you'll find helpful.
Consider the function $f$ defined by $x + 1$, when $x$ gets closer to $1$, $f$ gets closer to 2. 
But getting $x$ closer to 1 actually means that the difference between $x$ and $1$ is  as small as possible. Succinctly, it means 
  $0<|x - 1|<\delta$
Also, $f$ being made close to $2$ actually means we want
$0<|f(x) - 2|<\epsilon$ 
Here, epsilon and delta are positive real numbers that may or may not be less than 1( but as Spivak would put it, it's only of small epsilons that are of interest).
I only explained the absolute value terminology. You can visit YouTube and good textbooks(Mir Titles too) for further explanations.
A: One way of thinking about limit is imagining you have measuring devices for $x$ and $y$. A limit means that given any measuring device for $y$ with a particular amount of precision, there is a measuring device for $x$ with a precision for which for every $x$ close to $x_0$, $y$ is close to $y_0$. For instance, suppose $x$ is time and $y$ is distance. Then you have a watch for $x$ and a measuring tape for $y$. Each of those is going to have some degree of precision. For instance, the watch might measure times to a hundredth of a second, and the measuring tape measures distances to a millimeter. And maybe for every $x$ within a hundredth of second within $x_0$, $y$ is within a millimeter of $y_0$. Then we can say that for a watch with precision .01 s and tape with precision 1mm, if $x$ is close to $x_0$ then $y$ is close to $y_0$.
Saying that $f(x)$ goes to $y_0$ as $x$ goes to $x_0$ means that given any measuring tape $T$, there is some watch $W$ such that whenever $W$ measures $x$ as being indistinguishable from $x_0$ (that is, the difference between $x$ and $x_0$ is less than the precision of $W$), $T$ measures $y$ as being indistinguishable from $y_0$.
