In the preceding sections we have frequently been concerned with limits such as \[\lim_{n \to \infty} \phi_{n}(x),\] and series such as \[u_{1}(x) + u_{2}(x) + \dots = \lim_{n \to \infty}\{u_{1}(x) + u_{2}(x) + \dots + u_{n}(x)\},\] in which the function of \(n\) whose limit we are seeking involves, besides \(n\), another variable \(x\). In such cases the limit is of course a function of \(x\). Thus in § 75 we encountered the function \[f(x) = \lim_{n \to \infty} n(\sqrt[n]{x} – 1):\] and the sum of the geometrical series \(1 + x + x^{2} + \dots\) is a function of \(x\), viz. the function which is equal to \(1/(1 – x)\) if \(-1 < x < 1\) and is undefined for all other values of \(x\).

Many of the apparently ‘arbitrary’ or ‘unnatural’ functions considered in Ch. II are capable of a simple representation of this kind, as will appear from the following examples.

Example XXXI

1. \(\phi_{n}(x) = x\). Here \(n\) does not appear at all in the expression of \(\phi_{n}(x)\), and \(\phi(x) = \lim\phi_{n}(x) = x\) for all values of \(x\).

2. \(\phi_{n}(x) = x/n\). Here \(\phi(x) = \lim\phi_{n}(x) = 0\) for all values of \(x\).

3. \(\phi_{n}(x) = nx\). If \(x > 0\), \(\phi_{n}(x) \to +\infty\); if \(x < 0\), \(\phi_{n}(x) \to -\infty\): only when \(x = 0\) has \(\phi_{n}(x)\) a limit (viz. \(0\)) as \(n \to \infty\). Thus \(\phi(x) = 0\) when \(x = 0\) and is not defined for any other value of \(x\).

4. \(\phi_{n}(x) = 1/nx\), \(nx/(nx + 1)\).

5. \(\phi_{n}(x) = x^{n}\). Here \(\phi(x) = 0\), (\(-1 < x < 1\)); \(\phi(x) = 1\), (\(x = 1\)); and \(\phi(x)\) is not defined for any other value of \(x\).

6. \(\phi_{n}(x) = x^{n}(1 – x)\). Here \(\phi(x)\) differs from the \(\phi(x)\) of Ex. 5 in that it has the value \(0\) when \(x = 1\).

7. \(\phi_{n}(x) = x^{n}/n\). Here \(\phi(x)\) differs from the \(\phi(x)\) of Ex. 6 in that it has the value \(0\) when \(x = -1\) as well as when \(x = 1\).

8. \(\phi_{n}(x) = x^{n}/(x^{n} + 1)\). [\(\phi(x) = 0\), (\(-1 < x < 1\)); \(\phi(x) = \frac{1}{2}\), (\(x = 1\)); \(\phi(x) = 1\), (\(x < -1\) or \(x > 1\)); and \(\phi(x)\) is not defined when \(x = -1\).]

9. \(\phi_{n}(x) = x^{n}/(x^{n} – 1)\), \(1/(x^{n} + 1)\), \(1/(x^{n} – 1)\), \(1/(x^{n} + x^{-n})\), \(1/(x^{n} – x^{-n})\).

10. \(\phi_{n}(x) = (x^{n} – 1)/(x^{n} + 1)\), \((nx^{n} – 1)/(nx^{n} + 1)\), \((x^{n} – n)/(x^{n} + n)\). [In the first case \(\phi(x) = 1\) when \(|x| > 1\), \(\phi(x) = -1\) when \(|x| < 1\), \(\phi(x) = 0\) when \(x = 1\) and \(\phi(x)\) is not defined when \(x = -1\). The second and third functions differ from the first in that they are defined both when \(x = 1\) and when \(x = -1\): the second has the value \(1\) and the third the value \(-1\) for both these values of \(x\).]

11. Construct an example in which \(\phi(x) = 1\), (\(|x| > 1\)); \(\phi(x) = -1\), (\(|x| < 1\)); and \(\phi(x) = 0\), (\(x = 1\) and \(x = -1\)).

12. \(\phi_{n}(x) = x\{(x^{2n} – 1)/(x^{2n} + 1)\}^{2}\), \(n/(x^{n} + x^{-n} + n)\).

13. \(\phi_{n}(x) = \{x^{n}f(x) + g(x)\}/(x^{n} + 1)\). [Here \(\phi(x) = f(x)\), (\(|x| > 1\)); \(\phi(x) = g(x)\), (\(|x| < 1\)); \(\phi(x) = \frac{1}{2}\{f(x) + g(x)\}\), (\(x = 1\)); and \(\phi(x)\) is undefined when \(x = -1\).]

14. \(\phi_{n}(x) = (2/\pi) \arctan(nx)\). [\(\phi(x) = 1\), (\(x > 0\)); \(\phi(x) = 0\), (\(x = 0\)); \(\phi(x) = -1\), (\(x < 0\)). This function is important in the Theory of Numbers, and is usually denoted by \(\operatorname{sgn} x\).]

15. \(\phi_{n}(x) = \sin nx\pi\). [\(\phi(x) = 0\) when \(x\) is an integer; and \(\phi(x)\) is otherwise undefined (Ex. XXIV. 7).]

16. If \(\phi_{n}(x) = \sin (n!\, x\pi)\) then \(\phi(x) = 0\) for all rational values of \(x\) (Ex. XXIV. 14). [The consideration of irrational values presents greater difficulties.]

17. \(\phi_{n}(x) = (\cos^{2} x\pi)^{n}\). [\(\phi(x) = 0\) except when \(x\) is integral, when \(\phi(x) = 1\).]

18. If \(N \geq 1752\) then the number of days in the year \(N\) a.d. is \[\lim \{365 + (\cos^{2} \tfrac{1}{4} N\pi)^{n} – (\cos^{2} \tfrac{1}{100} N\pi)^{n} + (\cos^{2} \tfrac{1}{400} N\pi)^{n}\}.\]


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