To understand why we need the formal definition of a limit, you really have to understand why we need careful definitions at all in mathematics. A formal definition is really needed when intuition is insufficiently precise to avoid not only inconsistencies, but more importantly, the depth of understanding that allows us to see counterexamples to the intuitive definition that we would otherwise be blind to.
A very good example for the beginning calculus student is the relationship between continuity and the existence of the derivative-2 critical concepts in calculus you may or may not be yet aware of. Intuitively, a continuous function is one where there are no gaps in the graph of the function-that is,the function is defined on the entire domain. For example, the function f(x) = ax+b, where a,b are real numbers, is continuous. f has values defined on the entire real line-it defines a straight line in the plane. Where a= 2 and b=5,the graph is:
Now let's consider the function f(x) = 1/x. This function is not continous at x=0 and it's graph looks like this:
Clearly,there's a "hole" in the graph at x=0. But as you can also see,the graph of the function otherwise has no holes or gaps. If we exclude x = 0 from the domain of the function, (1/x) is continuous on it's domain by our "intuitive" definition. This is important,keep it in mind.
The related concept of the derivative is a measure of the rate of change of a function at a specific point, geometrically actualized by a tangent line to the curve at a specific point x= a at f(a).For example, given the function f(x) = x^2, the derivative is 2x and geometrically this gives:
Notice the derivative is a local property of f-i.e. it's defined at a point only. At most points on a continuous curve in the plane,it should be possible to construct a tangent line. So intitively, you would think that a function which was continuous would be differentiable i.e. the derivative would exist-at most points. Indeed, most mathematicians before the 19th century didn't worry about the existence of a derivative as long as the function's graph was sufficiently "smooth". So imagine the bewilderment in the 19th century when B.Bolzano and Karl Weierstrass both produced examples of functions which were everywhere continous on thier domains-and nowhere differentiable! To make the point, here's a computer generated graph of the Weierstrass function:
This is a function most mathematicians wouldn't have thought was possible given the "intuitive" definitions and machinery they'd been working with since Newton. If you think these are merely mathematical abstractions that have no relevance in the "real" world, you'd be very wrong indeed-it turns out many nonlinear phenomena, such as turbulence, exhibit precisely these kinds of mathematical behaviors locally! So it really is critical to have precision in mathematics to prevent our intution from leadong us astray. Intuition combined with rigor, in my opinion, is the most powerful human method of generating both mathematical and scientific facts.
That answer your question?