What is the epsilon-delta definition of limits, exactly? I am a bit confused with infinitesimals, and want to know why they were discarded and the epsilon-delta definition is being used? What is the epsilon-delta definition of limit? What is the intuition of limit in this definition? In the epsilon-delta definition, how do we find the limit? What is the difference between the infinitesimal approach and the epsilon-delta approach? 
 A: As far as I know, infinitesimals were not used in definitions of limit, but only in definitions of derivative and integral.
The history is (as to my knowledge) roughly as follows: calculus was "invented" by Newton and Leibniz, which have been concerned with the concepts of derivative and integral. However, in their treatment of these concepts, they have used a notion of infinitesimal (something like infinitely small) numbers, which failed to be rigorous (at least by modern standards).
However, Cauchy and others in the 19th century managed to make infinitesimal calculus rigorous by introducing the concept of the limit, which has been defined in the $\varepsilon$-$\delta$ notation. That is, the limit has (as far as I know)  always been defined via the $\varepsilon$-$\delta$ notation (perhaps except some early intuitive uses). For more information, see, e.g., Wikipedia.
There is also an another alternative approach to making calculus rigorous: Non-standard analysis. This roughly means that the original Newton's and Leibniz's definitions of derivative and integral are not totally abandoned and replaced by $\varepsilon$-$\delta$, but instead an axiomatic theory of infinitesimals is being built. However, this is quite advanced mathematics and the $\varepsilon$-$\delta$ approach is nowadays considered to be the mainstream.
However, I have to notice that I am not an expert on the history of calculus, so I may be wrong somewhere.
A: Infinitesimal approach is kinda out-dated in today's mathematics. The reason is that it's not rigorous enough, also, the advantage of the epsilon-delta definition is that it can be generalized to more general spaces. You know, before the progress of topology, mathematicians extensively used the idea behind the distance (Euclidean distance) between two points, which is called the metric function. I mean, when they wanted to think about closeness of two points in a space, they thought about the distance between those two points. The notion of 'closeness'(being close) is now a bit more generalized in today's mathematics and it has been replaced by open sets and neighborhoods.  Later they realized that many properties of the real line and many concepts in mathematical analysis could be independently studied in a more general approach which doesn't necessarily use the idea of a metric function for determining how close two things are. The epsilon-delta definition of continuity has the advantage that it could be seen as an intermediate step in generalizing the concept of a continuous map to more general spaces where you can't define a metric function.
The epsilon-delta definition of limit says that if you want f(x) to be arbitrary close to a value L as x approaches a, i.e $\displaystyle lim_{x \to a} f(x)= L$, all you need to do as that you find a delta such that the distance between x and a is smaller than delta. We can write this in math symbols as:
$\displaystyle \forall \epsilon, \exists \delta$ s.t. $0<|x-a|<\delta$ $\implies$ $|f(x)-L|<\epsilon$
For continuity, you allow the case x=a, so the definition becomes 
$\displaystyle \forall \epsilon, \exists \delta$ s.t. $|x-a|<\delta$ $\implies$ $|f(x)-f(a)|<\epsilon$
Notice that you can talk about $f(a)$ instead of $L$ since f is continuous.
This intuitively means that you can make f(x) arbitrarily close as you want to L (or f(x)) provided that you take x close enough to a. What does close enough to a mean in this case? It means that the distance between x and a shouldn't exceed delta. This is what it means. If you take x close enough to a, then f(x) will be close enough to L (or f(a)). And here, close enough means that the distance between f(x) and L (or f(a)) is less than your desired epsilon.
I hope that it's clear now.
A: First off, what do you mean by infinitesimal? Can you define it in a mathematically rigorous way? People tried for hundreds of years but nobody could (until the 20th century notion of "hyperreals", but nobody actually uses these). So we had to sacrifice intuition for rigor, which led to the epsilon-delta definition. One way to get a better feel for the epsilon-delta definition, is that if the limit of a function $f(x)$ as $x$ approaches $a$ is the number $L$, this means that given any interval $J$ you may find that contains $L$ (meaning an open subset of the real line of the form $J=(s,t)$), there is an interval $I$ containing $a$ such that $f(I)$ is contained in $J$. (Draw a picture associated with what I just stated to make things even more apparent.)  
