Limit Question - Explanation The limit of $f(x) = x$, as $x$ tends to zero is zero.
What's the limit of the function $\dfrac{x^2}{x}$ as $x$ tends to zero? and
What's the limit of the function f(x) = (modulus of x)/x ? 
I am interested in this question from a teaching perspective. How would you explain and contrast these three cases in an intuitive manner to a student?(i.e something like " if you take values of $x$ arbitrary close to zero then .....or using the epsilon-delta definition in an intuitive way.) I know that the epsilon-delta definition is not exactly intuitive, but I guess, a reasonable degree of intuition should be possible for such simple cases.
I am not a teacher, but I think you understand a concept properly only when you are able to explain it clearly to someone else, so....
 A: This has already been discussed quite a bit, but since my understanding of "explain ... in an intuitive manner to a student" seems to be quite different from the discussion so far, I thought I'd offer my approach to this.
The basic idea is to write $x^2$ as $x \cdot x$, which allows us to see that $x^2$ is a small number times $x$ (when $x$ is small). For instance:
When $x = \frac{1}{10},$ then $x \cdot x = \frac{1}{10} x,$ and hence $x^2$ is $10$% of $x.$
When $x = \frac{1}{100},$ then $x \cdot x = \frac{1}{100} x,$ and hence $x^2$ is $1$% of $x.$
When $x = \frac{1}{1000},$ then $x \cdot x = \frac{1}{1000} x,$ and hence $x^2$ is $0.1$% of $x.$
When $x = \frac{1}{10000},$ then $x \cdot x = \frac{1}{10000} x,$ and hence $x^2$ is $0.01$% of $x.$
As $x$ gets smaller and smaller, the values of $x^2$ become a smaller and smaller percent of the values of $x.$ Algebrically, we can see this easily from $\frac{x^2}{x} = x,$ but I think most anyone who is initially puzzled over what is going on intuitively will probably not be helped much by someone immediately pointing out the identity $\frac{x^2}{x} = x.$ However, with the examples above in sight, such a student can now see that $\frac{x^2}{x} = x$ efficiently summarizes the results seen in these examples.
Naturally, at some point we should point out what we really mean by "smaller and smaller". For example, the values of $1 + \frac{1}{n}$ for $n=1,\;2,\ldots$ get smaller and smaller. But, for an initial investigation into what Nikhil Panikkar asked about, these are things that can be looked at later (along with the idea of having to use arbitrary sequences approaching $0,$ the use of epsilon-delta neighborhood language, etc.).
A: I think a concept that this question is trying to elucidate is that the limit depends on values of a function near a point rather than at a point. Specifically: $\frac{x^2}{x}$ is equal to $x$ everywhere except at $x = 0$, where the former function is undefined. But for the purposes of taking a limit as $x \rightarrow 0$, the behavior at $x = 0$ doesn't matter.
Similarly, you might ask about the limit as $x \rightarrow 0$ of a function:
$$f(x) = \left\{\begin{array}{cc} 50 & x = 0 \\ x & x \neq 0 \end{array}\right.$$
Of course, this functions isn't equal to $x$, but when it comes to taking a limit that doesn't matter.
A: Let me start my answer rigorously.
Lemma: Let $x_0$ be a point and $f,g$ be two functions and supposed that at a punctuated neighborhood of $x_0$ the functions equals. Then $\lim_{x\to x_0} f(x)=\lim_{x\to x_0} g(x)$. 
Thus, since $x^2/x$ and $x$ equals in the punctured neighborhood $0<|x|<1$ of $x_0=0$, they have the same limit. 
The intuition that should be explained to a student is that 
1) limit of a function is local (depends on the values only near the point $x_0$)
2) the limit does not depend on the function at the actual points (so depends only on values in the punctured neighborhood...)
I think a student may well understand these two points intuitively, and hence be able to understand why $x$ and $x^2/x$ have the same limit as $x\to 0$. 
