AN INTRODUCTION TO CALCULUS THE LIMIT OF A FUNCTION In the previous section, we examined tangents and how to compute their equations. The examples we treated showed that limits arise whenever we attempt to compute velocities of objects, or tangents to curves. In the first example of the previous section, recall that we dealt with the reciprocal function y = 1/x, and tried to compute the equation of the tangent line at point P (0.5, 2). From the tables, we saw that, as x approached 0.5 from either side (that is, from the left and right of 0.5), the values of the slopes of the secant lines approached –4. Thus, we assumed that the slope of the tangent line at P was –4. Consequently, this allowed us to compute the equation of the tangent. From the example and the others, it is clear that the slope of a tangent is the limit of the slope of the secant lines . Assuming we represent the slope of a typical secant line by MPQ, and the slope of the tangent by M, then,symbolically, the situation is expressed like this:
lim MPQ
Q→P
=
M
1
This mathematical representation will be explained in detail as we move on. Similarly, for the velocity problems, we saw that the instantaneous velocity of an object is the limit of average velocities . Here's one way of defining a limit:
Given two quantities x and f(x), we can take x to be sufficiently close to a number a (as close as possible, but not equal to a), so that the values of f(x) will be as close to another number L as possible). And so, we write
lim
x→a
f(x)
=
L
2
which is read as “the limit of f(x) , as x approaches a, is L”.
Thus, equation 1 above can be read as
“the limit of the slope of the secant line MPQ, as Q(x, y) approaches P(0.5, 2), equals the slope of the tangent, M”. In the previous, I mentioned that the idea of one quantity f(x) approaching a value L, as another related quantity x approaches a value a, is the most basic concept of a limit.
Greek scholars like Archimedes and Eudoxus used a “method of exhaustion” to calculate areas and volumes, a method in which limits were indirectly used. Even when calculus was in its developmental stages, Newton's immediate predecessors (like Barrow, Fermat and Cavalieri), did not use limits. Sir Isaac Newton, who was the first mathematician to speak explicitly about limits explained (and I quote): “quantities approach nearer than by any given difference”. He further stated that the limit was the basic concept in calculus, but such ideas were left to later mathematicians like Cauchy to clarify. Study this example carefully.
EXAMPLE 1 Assume you have the function
g(x) =
x–1 x3 – 1