184 
MISCELLANEOUS EXAMPLES ON CHAPTER Y 
say. It is evident that r] is of the second order of smallness, xr] of the third, 
and rj 2 of the fourth ; and -2erj = Ax 2 + 2ABX 3 , the error being of the fourth 
order.] 
10. If x=ay-\-by 2 +cy z , then one value of y is given by 
y = ax+/3a- 2 + yx 3 (1 + e x ), 
where a = l/a, /3 — - 6/a 3 , y = (26 2 —ac)/a 5 , and e x is of the first order of small 
ness when x is small. 
11. If x=ay + by n , where n is an integer greater than unity, then one 
value of y is given by y=ax + ¡3x n +yx 2n “ 1 (1 -he*), where a = 1 /a, /3 = — b/a n + 1 , 
y=nb 2 /a 2n + 1 , and e x is of the {n — l)th order of smallness when x is small. 
12. Prove that (sec x - tan x) -*■ 0 as x 
13. Prove that fy(x) = 1 — cos (1 - cos x) is of the fourth order of smallness 
when x is small; and find the limit of <p (x)jx i as 0. 
14. Prove that (/) {x)=#sin (sin#) — sin 2 # is of the sixth order of smallness 
when x is small; and find the limit of <p (x)/x‘ ] as #-*•(). 
15. From a point T in a radius OA of a circle produced, a tangent TP is 
drawn to the circle, touching it in P, and PN is drawn perpendicular to OA. 
Show that JVA/A7 1 -*! as P moves up to A. 
16. Tangents are drawn to a circular arc at its middle point and its 
extremities : A is the area of the triangle formed by the chord of the arc and 
the two tangents at the extremities, and A' the area of that formed by the 
three tangents. Show that, as the length of the arc tends to zero, A/A'-»-4. 
IT. For what values of a does {a+sin (1/#)}/# tend to (1) -foe , (2) - oo, 
as x-*~01 [To -foo if a>l, to - oo if a< —1: if -l^a<l the function 
oscillates.] 
18. For what values of x is the function </>(#) = ! - lim ^/(cos 2 tr#) con- 
n-^-cc 
tinuous or discontinuous ? 
19. Show that the least positive root of the equation sin#=a# is a 
continuous function of a throughout the interval (0, 1), and increases steadily 
from 0 to n as a decreases from 1 to 0. [The function is the inverse of 
(sin x)/x: apply § 88.] 
20. The least positive root of tan x=ax is a continuous function of a 
throughout the interval (1, oo), and increases steadily from 0 to %rr as a 
increases from 1 towards qo . 
21. If (p (x)=-l ¡q when x=p/q, (x) — 0 when x is irrational, cp(x) is 
continuous for all irrational and discontinuous for all rational values of x. 
22. Let </> (x)=x when x is rational and cp{x)~l -x when x is irrational. 
Show that, as x increases from 0 to 1, </> (,r) assumes every value, between 
and including 0 and 1, once and once only, although </> (x) is discontinuous 
for every value of x except x=