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 $\lim \{365 + (\cos^{2} \tfrac{1}{4} N\pi)^{n} – (\cos^{2} \tfrac{1}{100} N\pi)^{n} + (\cos^{2} \tfrac{1}{400} N\pi)^{n}\}.$