150 LIMITS OF FUNCTIONS OF AN INTEGRAL VARIABLE [iV 
Examples XXXIIL 1. (p n (x)=x. Here n does not appear at all in the 
expression of (p n (x), and <p (x) = lirn (p H {x) =x for all values of x. 
2. 4>u {x)=x/n. Here <p{x) = \im (p n (x)=0 for all values of x. 
3. <p n {x) = nx. If #>0, (p n {x)^~ + oo ; if x<0, <p H (x)-*■ - oo : only for 
A’=0has (p n {x) a limit (viz. 0) as n-*~oo. Thus (p{x) = 0 when x=0 and is 
not defined for any other value of x. 
4. (p n {x) = ljnx, nx/{nx+l). 
5. (p n {x)=x n . Here (p(x) =0, ( - l<a’<l); (p (x) = l, (x = l); and (p(x) 
is not defined for any other value of x. 
6. (p H {x)=x n (1 —x). Here <p (x) differs from the <p(x) of Ex. 5 in that 
it is defined and has the value 0 for x—1. 
7. (p n {x)=x n /n. Here <p (x) differs from the <p(x) of Ex. 6 in that it is 
defined and has the value 0 for x= -1 as well as +1. 
8. <p n (x)=tf n /(x n +1). [(p{x)=0, (~l<x<l); (p{x)=h, (x—l); (p{x) = l, 
(x< — 1 or jt>1) ; and (p (x) is not defined for x — — 1.] 
9. (p n (x)=x n /{x n - 1), l/(A' n +l), l/(x n — 1), 1 l(x n +x~ n ), l/{x n -x~ n ). 
10. (p n (x)=(x n -l)/{x n + l), (nx n — l)/(nx n +l), (x n -n)/{x H + n). [In the 
first case (p{x) = 1 if ||> 1, (p{x) = -1 if U|<1, (p{x)=0 if a’=1 and (p{x) 
is not defined for x— — 1. The second and third functions differ from the 
first in that they are defined both for x=\ and x= -1: the second has the 
value 1 and the third the value - 1 for both these values of xJ] 
11. Construct an example in which <£(a’) = 1, (|a’|>1); cp(x)=-1, 
(| a? | < 1); and (p{x)=0, (#=±1). 
12. cp n {x)=x{{x 2n -l)j{x 2,l +l)} 2 , n/(x n + x~ H + n). 
13. (p n (x) = {x n f(x) +g(x)}/(X n +1). [Here <p{x)=f(x), (|#|>1); (p (x) = 
g{x\ (|#|<1) ; (P{x) = %{f{x)+g{x)}, (#=1); and cp {x) is undefined for 
x=-l.\ 
14. (p n (x) = {2ln)a,Ycta,n{nx). [<p{x) = l, (x>0); <p (x) = 0, (x=0); 
(p{x)= — 1, (^<0). This function is important in the Theory of Numbers, 
and is usually denoted by sgn xJ\ 
15. <p n (x) = {l/n) sin nxn. [cp {x)—0 for all values of x.~\ 
16. (p n {x)—sin nxrr. [cp (x) — 0 when x is an integer, and is otherwise 
undefined.] 
17. (p n (x)={ljn) cos nxjr, cos nxrr, a cos 2 nxir + h sin 2 nxre. 
18. If (p n {x) = sin{n\x7r), cp{x)=0 for all rational values of x (Exs. XXVI. 9, 
XXVII. 8). The consideration of irrational values presents greater difficulties. 
19. (p n (x) = (cos 2 a'tt)’ 1 . [(p(x)=0 except when x is integral, when 
4>(x)=i.] 
20. (p n {x) = (sin 2 X7r') H ) (cosxn) n , (sin#«-)* 1 .