The (ε, δ) definition of limit , is it essential ? What is its importance ? and the difference between it and normal limit calculation? As the title says , why should we learn (ε, δ) definition of limit ? is it essential ? What is its importance ? and the difference between it and normal limit evaluation?
 A: The “$\varepsilon$-$\delta$ definition” is what allows you to do “normal evaluation”, for example
$$
\lim_{x\to0}\frac{1-\cos x+\sin x}{x}
=
\lim_{x\to0}\frac{1-\cos x}{x}+\lim_{x\to0}\frac{\sin x}{x}=0+1=1
$$
because the theorem about sums of limits is proved by means of the “$\varepsilon$-$\delta$ definition”.
Otherwise such a computation would be meaningless or simply refer to an intuition that revealed faulty a couple of centuries ago.
A: Consider this example $\lim_{x \to \infty} e^{-x} = 0$. The problem with this calculation is that one may be confused as the to think that the value of $e^{-x}$ is zero as $x$ becomes large. 
What we learn from the $\delta$-$\epsilon$ definition of the limit which states for $\epsilon > 0$ there exists a $\delta >0$ such that whenever $0 < | x-c| < \delta$ this implies that $|f(x) - f(c)| < \epsilon$, that when we have this condition satisfied (i.e $0<|x-c|<\delta$) in fact we realize that the limit is not the value of the function at that point, but the value of the function $f(x)$ becomes infinitely close to the value $f(c)$ as $x$ approaches $c$. 
Moreover, mathematical rigor is essential in the way we end up generalizing mathematical theorems. Hope this helps. 
A: It's kind of like when, at one point in their education, one learns that, for example, we can treat problems in an abstract way by representing numbers as symbols. This allowed us to solve a whole new gamma of problems by representing the supposed solution as, say, $x$, and then manipulating by means of multiplication, addition, and other operations which we previously only viewed as applicable in themselves, e.g. for saying things like $3\times 4 = 12$ (maybe, at this point of abstraction, someone would have said that $3\times 4=12$ is "normal multiplication", and didn't see the need for using an abstract language the likes of $x$,$y$, and $x\cdot y $). The situation is a bit similar here: you ask why not just use "normal limit evaluation" because after all, we just need to evaluate expressions like $$\lim_{x\to0}e^x$$
But really, one needs to be able to deal with limits in an abstract way to solve further problems, just as before we needed to introduce symbols for representing numbers, to solve a wide array of newer, more abstract problems. Note: I'm not saying that this is in any way the historical development of things, but it may be the way we develop it in our minds, possibly...
Of course I'll need to add an example where the abstract conception of the "limit definition" is necessary. L'Hôpital comes to mind, as well as Taylor expansions and other theorems. But take something simple, for example, how can we be sure that $$\lim_{x\to a}\ln f(x)=\ln \lim_{x\to a}f(x) \quad ?$$
clearly we need definitions of limits and continuity, rather than just knowing how to evaluate specific limits. Did that explain it a bit?
