Limit of a variable

If a variable $v$ takes on successively a series of values that approach nearer and nearer to a constant value $L$ in such a manner that $\vert v - L \vert$ becomes and remains less than any assigned arbitrarily small positive quantity, then $v$ is said to approach the limit $L$, or to converge to the limit $L$. Symbolically this is written

$$\lim_{v = L},\ \ {\rm or},\ \ \lim_{v \rightarrow L}.$$

The following familiar examples illustrate what is meant:

1. As the number of sides of a regular inscribed polygon is indefinitely increased, the limit of the area of the polygon is the area of the circle. In this case the variable is always less than its limit.

2. Similarly, the limit of the area of the circumscribed polygon is also the area of the circle, but now the variable is always greater than its limit.

3. Consider the series

   $$1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots$$ (3.1)

   
   
   The sum of any even number $(2n)$ of the first terms of this series is

   $$\begin{array}{ll} S_{2n} &= 1 - \frac{1}{2} + \frac{1}{4} - \... ...} \\ &= \frac{2}{3} - \frac{1}{3 \cdot 2^{2n - 1}}, \end{array}$$ (3.2)

   
   
   by item 6, Ch. 1, §1.1. Similarly, the sum of any odd number $(2n + 1)$ of the first terms of the series is

   $$\begin{array}{ll} S_{2n + 1} & = 1 - \frac{1}{2} + \frac{1}{4... ... - 1} \\ &= \frac{2}{3} + \frac{1}{3 \cdot 2^{2n}}, \end{array}$$ (3.3)

   
   
   again by item 6, Ch. 1, §1.1.

   Writing (3.2) and (3.3) in the forms

   $$\frac{2}{3} - S_{2n} = \frac{1}{3 \cdot 2^{2n - 1}}, \ \ \ \ \ S_{2n + 1} - \frac{2}{3} = \frac{1}{3 \cdot 2^{2n}}$$
   we have
   $$\lim_{n \to \infty} \left ( \frac{2}{3} - S_{2n} \right ) = \lim_{n \to \infty} \frac{1}{3 \cdot 2^{2n - 1}} = 0,$$
   and
   $$\lim_{n \to \infty} \left ( S_{2n + 1} - \frac{2}{3} \right ) = \lim_{n \to \infty} \frac{1}{3 \cdot 2^{2n}} = 0.$$
   Hence, by definition of the limit of a variable, it is seen that both $S_{2n}$ and $S_{2n + 1}$ are variables approaching $\frac{2}{3}$ as a limit as the number of terms increases without limit.

   Summing up the first two, three, four, etc., terms of (3.1), the sums are found by ((3.2) and ((3.3) to be alternately less and greater than $\frac{2}{3}$, illustrating the case when the variable, in this case the sum of the terms of ((3.1), is alternately less and greater than its limit.

In the examples shown the variable never reaches its limit. This is not by any means always the case, for from the definition of the limit of a variable it is clear that the essence of the definition is simply that the numerical value of the difference between the variable and its limit shall ultimately become and remain less than any positive number we may choose, however small.

Example 3.1.1   As an example illustrating the fact that the variable may reach its limit, consider the following. Let a series of regular polygons be inscribed in a circle, the number of sides increasing indefinitely. Choosing anyone of these, construct. the circumscribed polygon whose sides touch the circle at the vertices of the inscribed polygon. Let $p_n$ and $P_n$ be the perimeters of the inscribed and circumscribed polygons of $n$ sides, and $C$ the circumference of the circle, and suppose the values of a variable $x$ to be as follows:

$$P_n,\ \ p_{n + 1},\ \ C,\ \ P_{n + 1},\ \ p_{n + 2},\ \ C,\ \ P_{n + 2}, \ \ \ \ {\rm etc.}$$

Then, evidently,

$$\lim_{x \to \infty} x = C$$

and the limit is reached by the variable, every third value of the variable being $C$.