And ? would not be the upper bound of the class

Info icon This preview shows pages 249–252. Sign up to view the full content.

, 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.
Image of page 249

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

[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.
Image of page 250
[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 ] .
Image of page 251

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 252
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern