The concept of a limit is perhaps the most difficult one in calculus. Intuitively, the idea of a limit is not hard to grasp, but the rigorous definition requires some attention. Formalizing geometric intuition is one of the key skills in studying analysis, a branch of mathematics that generalizes many of the concepts in basic calculus. This post will attempt to show that the rigorous definition of the limit is reasonable and does not impede the intuitive notion of a limit.
Let us start by acquiring the intuition behind limits. When we say that a function f approaches a limit L near a, we mean that we can make f(x) as close to L as we want by requiring that x be sufficiently close to, but unequal to, a point a. It is clear from this “definition” that we do not care about the value of f at a, nor do we need to know whether f is defined at a. We simply want to see if f(x) gets closer and closer to L whenever x is arbitrarily close to a. However, it is hardly clear what we mean by requiring f(x) to be “close to” L, or x “close to” a. After we gain a geometric sense of these ideas, we can begin formalizing each detail.
In the figure below, we have a function f and open intervals on the y-axis and x-axis. If we want f(x) to be close to L within the open interval B centered at L, then we need the open interval A centered at a to contain points that are within the region bounded by the lines through the endpoints of B. In (a), we have picked an open interval A that works (we could have chosen an even larger open interval A). But the key idea behind the limit is that if we choose a smaller open interval B’ centered at L and ask that f(x) to be close to L within that interval, we can always find an open interval A’ (usually smaller than A) that guarantees this. Clearly, the open interval A in (b) is too large to guarantee that f(x) is even within the open interval B, let alone in a smaller open interval B’. We can now translate our geometric ideas into precise mathematical terms.
The best way to ease into the formal definition of limits is through an example. Consider the function (|x|)^(1/2) * sin(1/x), whose graph is below.
The graph shows that f appears to be approaching 0 near 0. Let us see how far our intuitive “definition” can get us. In this case, both L and a are 0, and we want to see if we can get f(x) = (|x|)^(1/2) * sin(1/x) as close to 0 as we want by requiring that x be sufficiently close to, but unequal to, 0. Suppose we want (|x|)^(1/2) * sin(1/x) to be within 1/100 of 0. This means that we want an open interval of radius 1/100 centered at 0, or -1/100 < (|x|)^(1/2) * sin(1/x) < 1/100. But this last chain of inequalities is equivalent to |(|x|)^(1/2) * sin(1/x)| < 1/100 (recall that |a – b| denotes the distance between points a and b). Since sin(1/x) =< 1, for all x =/= 0,
(1) |(|x|)^(1/2) * sin(1/x)| = |(|x|)^(1/2)| * |sin(1/x)| =< |(|x|)^(1/2)| = |x|^(1/2).
Now |x| = |x – 0| signifies the distance from x to 0. If we want | (|x|)^(1/2) * sin(1/x) | < 1/100 and we have | (|x|)^(1/2) * sin(1/x) | =< |x|^(1/2) by (1), then clearly letting |x| < 1/10000 and x =/= 0 works because then |x|^(1/2) < 1/100 and x =/= 0, implying | (|x|)^(1/2) * sin(1/x) | =< |x|^(1/2) < 1/100 for all x =/= 0. Thus, we have shown that (|x|)^(1/2) * sin(1/x) is within 1/100 of 0 provided that x is within 1/10000 of 0, but =/= 0.
But this only solves the specific case where we desire f(x) to be within 1/100 of 0. We also wish to show that for a smaller open interval centered at 0 on the y-axis, we can find an open interval on the x-axis that works. There is nothing special about 1/100 and instead we could take any number E > 0 and make |f(x) – 0| < E by requiring that |x| < E^2 and x =/= 0 (work this generalized case out on your own… why choose E^2 ?).
Although one example might not be enough, we can already find fault with our “definition”. Let us take a stab at making the intuitive idea more precise. Recall our intuitive “definition”: the function f approaches a limit L near a, if we can make f(x) as close to L as we want by requiring that x be sufficiently close to, but unequal to, a. The first change to make is to realize that making f(x) close to L means making |f(x) – L| small, and similarly for x and a. Therefore, our definition becomes, the function f approaches a limit L near a, if we can make |f(x) – L| as small as we want by requiring that |x – a| to be sufficiently small, and x =/= a.
Next, we know that making |f(x) – L| as small as we want means making |f(x) – L| <> 0, so our definition becomes the function f approaches a limit L near a, if given any number E > 0 we can make |f(x) – L| < E by requiring that |x – a| to be sufficiently small, and x =/= a.
Finally, we arrive at the crucial step. Our example above shows us that for each number E > 0, we found a different number D > 0 such that if x =/= a and |x – a| < D, then |f(x) – L| < E. Specifically, this number D in our example was E^2 (usually D is a function of E but this is not always the case). We finally have our precise definition:
The function f approaches a limit L near a, if given any number E > 0 there is some D > 0 such that, for all x, if |x – a| < D and x =/= a, then |f(x) – L| < E.
Note that requiring |x – a| < D and x =/= a might as well be denoted as 0 < |x – a| < D, so our final definition is given below.
Formal Definition of the Limit: The function f approaches the limit L near a means: given any E > 0, there exists some D > 0 such that, for all x, if 0 < |x – a| < D, then |f(x) – L| < E.
Type: art.discourse.mathematics
Produced by: The Numbers
Type: art.aural.original
