Intuitively, to say that the limit of a function f(x) as x approaches a is equal to L , just means that if x is sufficiently close to the value a, then f(x) is sufficiently close to the value L. Thus, for example, we say intuitivley that the limit of the function f(x) = 2x +1 as x approaches 2 is equal to 5, since if x is close to 2 then f(x) is close to 5.
This intuitive notion is easy to understand but unfortunately, it lacks mathematical rigor. Consider for example the limit as a x approaches 0 of the function f(x) = (sin x)/x. The first difficulty with this problem is that the function cannot be evaluated at 0 since division by 0 is not permitted. If we use a calculator (in radian mode) and compute the value of (sin x)/x for small values of x such as 0.1 or 0.01, we see the value of the function is close to 1, but it is not 1! As x becomes even closer to 0, (sin x)/x gets closer and closer to 1, although, it never really gets there.
Graph of the Function f(x)=(sin x)/x
The function has a "hole" at x = 0, but as the value of x gets closer and closer to 0 the value of the function gets closer and closer to 1.
When is "close" close enough?
What we would like to say somehow is that we can get arbitrarily close, but "arbitrarily close," is not a number. To get around this difficulty, Cauchy proposed a very rigorous definition of limits, in which, "arbitrarily close" is described in terms of an arbitrary number "epsilon."