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# And ? would not be the upper bound of the class

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, and η would not be the upper bound of the class considered. Hence F ( η ) = ξ . The equation F ( y ) = ξ has therefore a unique solution y = η = φ ( ξ ), say; and plainly η increases steadily and continuously with ξ , which proves the theorem. MISCELLANEOUS EXAMPLES ON CHAPTER V. 1. Show that, if neither a nor b is zero, then ax n + bx n - 1 + · · · + k = ax n (1 + x ) , where x is of the first order of smallness when x is large. 2. If P ( x ) = ax n + bx n - 1 + · · · + k , and a is not zero, then as x increases P ( x ) has ultimately the sign of a ; and so has P ( x + λ ) - P ( x ), where λ is any constant.

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[V : 109] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 234 3. Show that in general ( ax n + bx n - 1 + · · · + k ) / ( Ax n + Bx n - 1 + · · · + K ) = α + ( β/x )(1 + x ) , where α = a/A , β = ( bA - aB ) /A 2 , and x is of the first order of smallness when x is large. Indicate any exceptional cases. 4. Express ( ax 2 + bx + c ) / ( Ax 2 + Bx + C ) in the form α + ( β/x ) + ( γ/x 2 )(1 + x ) , where x is of the first order of smallness when x is large. 5. Show that lim x →∞ x { x + a - x } = 1 2 a. [Use the formula x + a - x = a/ { x + a + x } .] 6. Show that x + a = x + 1 2 ( a/ x )(1+ x ), where x is of the first order of smallness when x is large. 7. Find values of α and β such that ax 2 + 2 bx + c - αx - β has the limit zero as x → ∞ ; and prove that lim x { ax 2 + 2 bx + c - αx - β } = ( ac - b 2 ) / 2 a . 8. Evaluate lim x →∞ x q x 2 + p x 4 + 1 - x 2 . 9. Prove that (sec x - tan x ) 0 as x 1 2 π . 10. Prove that φ ( x ) = 1 - cos(1 - cos x ) is of the fourth order of smallness when x is small; and find the limit of φ ( x ) /x 4 as x 0. 11. Prove that φ ( x ) = x sin(sin x ) - sin 2 x is of the sixth order of smallness when x is small; and find the limit of φ ( x ) /x 6 as x 0. 12. From a point P on a radius OA of a circle, produced beyond the cir- cle, a tangent PT is drawn to the circle, touching it in T , and TN is drawn perpendicular to OA . Show that NA/AP 1 as P moves up to A . 13. Tangents are drawn to a circular arc at its middle point and its ex- tremities; Δ is the area of the triangle formed by the chord of the arc and the two tangents at the extremities, and Δ 0 the area of that formed by the three tangents. Show that Δ / Δ 0 4 as the length of the arc tends to zero.
[V : 109] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 235 14. For what values of a does { a + sin(1 /x ) } /x tend to (1) , (2) -∞ , as x 0? [To if a > 1, to -∞ if a < - 1: the function oscillates if - 1 5 a 5 1.] 15. If φ ( x ) = 1 /q when x = p/q , and φ ( x ) = 0 when x is irrational, then φ ( x ) is continuous for all irrational and discontinuous for all rational values of x . 16. Show that the function whose graph is drawn in Fig. 32 may be repre- sented by either of the formulae 1 - x + [ x ] - [1 - x ] , 1 - x - lim n →∞ (cos 2 n +1 πx ) . 17. Show that the function φ ( x ) which is equal to 0 when x = 0, to 1 2 - x when 0 < x < 1 2 , to 1 2 when x = 1 2 , to 3 2 - x when 1 2 < x < 1, and to 1 when x = 1, assumes every value between 0 and 1 once and once only as x increases from 0 to 1, but is discontinuous for x = 0, x = 1 2 , and x = 1. Show also that the function may be represented by the formula 1 2 - x - 1 2 [2 x ] - 1 2 [1 - 2 x ] .

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