TABLE OF CONTENTS Mathematicians base everything in calculus on two concepts: the limit of an infinite sequence, and the limit of a function. We have done the first. The second concept turns out to be easy to illustrate and easy to explain informally, but very difficult to define in a rigorous way.
We begin with some function of x, say simply f( x), and some constant, a, such that f( a) = L. If we can make the value of f( x) as close to L as we want by choosing values of x close enough to a but nonetheless different from a, then we say that the limit of f( x), as x approaches a, is L. We write this as
which we read as "the limit of f of x, as x approaches a, is L."
This means that if x is close to a, then f( x) is close to L. If another value of x is even closer to a, then the resulting f( x) is also even closer to L. If x ? a is small, then f( x) ? L should be small. When we say "small" we mean arbitrarily small or below any ?. The difference f( x) ? L
