1. Show that, if neither $$a$$ nor $$b$$ is zero, then $ax^{n} + bx^{n-1} + \dots + k = ax^{n} (1 + \epsilon_{x}),$ where $$\epsilon_{x}$$ is of the first order of smallness when $$x$$ is large.

2. If $$P(x) = ax^{n} + bx^{n-1} + \dots + k$$, and $$a$$ is not zero, then as $$x$$ increases $$P(x)$$ has ultimately the sign of $$a$$; and so has $$P(x + \lambda) – P(x)$$, where $$\lambda$$ is any constant.

3. Show that in general $(ax^{n} + bx^{n-1} + \dots + k)/(Ax^{n} + Bx^{n-1} + \dots + K) = \alpha + (\beta/x) (1 + \epsilon_{x}),$ where $$\alpha = a/A$$, $$\beta = (bA – aB)/A^{2}$$, and $$\epsilon_{x}$$ 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 $\alpha + (\beta/x) + (\gamma/x^{2})(1 + \epsilon_{x}),$ where $$\epsilon_{x}$$ is of the first order of smallness when $$x$$ is large.

5. Show that $\lim_{x\to\infty}\sqrt{x}\{\sqrt{x + a} – \sqrt{x}\} = \tfrac{1}{2} a.$

[Use the formula $$\sqrt{x + a} – \sqrt{x} = a/\{\sqrt{x + a} + \sqrt{x}\}$$.]

6. Show that $$\sqrt{x + a} = \sqrt{x} + \frac{1}{2}(a/\sqrt{x}) (1 + \epsilon_{x})$$, where $$\epsilon_{x}$$ is of the first order of smallness when $$x$$ is large.

7. Find values of $$\alpha$$ and $$\beta$$ such that $$\sqrt{a x^{2} + 2bx + c} – \alpha x – \beta$$ has the limit zero as $$x \to \infty$$; and prove that $$\lim x\{\sqrt{ax^{2} + 2bx + c} – \alpha x – \beta\} = (ac – b^{2})/2a$$.

8. Evaluate $\lim_{x \to\infty} x\left\{\sqrt{x^{2} + \sqrt{x^{4} + 1}} – x\sqrt{2}\right\}.$

9. Prove that $$(\sec x – \tan x) \to 0$$ as $$x \to \frac{1}{2}\pi$$.

10. Prove that $$\phi(x) = 1 – \cos(1 – \cos x)$$ is of the fourth order of smallness when $$x$$ is small; and find the limit of $$\phi(x)/x^{4}$$ as $$x \to 0$$.

11. Prove that $$\phi(x) = x\sin(\sin x) – \sin^{2}x$$ is of the sixth order of smallness when $$x$$ is small; and find the limit of $$\phi(x)/x^{6}$$ as $$x \to 0$$.

12. From a point $$P$$ on a radius $$OA$$ of a circle, produced beyond the circle, a tangent $$PT$$ is drawn to the circle, touching it in $$T$$, and $$TN$$ is drawn perpendicular to $$OA$$. Show that $$NA/AP \to 1$$ as $$P$$ moves up to $$A$$.

13. Tangents are drawn to a circular arc at its middle point and its extremities; $$\Delta$$ is the area of the triangle formed by the chord of the arc and the two tangents at the extremities, and $$\Delta’$$ the area of that formed by the three tangents. Show that $$\Delta/\Delta’ \to 4$$ as the length of the arc tends to zero.

14. For what values of $$a$$ does $$\{a + \sin(1/x)\}/x$$ tend to (1) $$\infty$$, (2) $$-\infty$$, as $$x \to 0$$? [To $$\infty$$ if $$a > 1$$, to $$-\infty$$ if $$a < -1$$: the function oscillates if $$-1 \leq a \leq 1$$.]

15. If $$\phi(x) = 1/q$$ when $$x = p/q$$, and $$\phi(x) = 0$$ when $$x$$ is irrational, then $$\phi(x)$$ 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 represented by either of the formulae $1 – x + [x] – [1 – x],\quad 1 – x – \lim_{n\to\infty} (\cos^{2n+1}\pi x).$

17. Show that the function $$\phi(x)$$ which is equal to $$0$$ when $$x = 0$$, to $$\frac{1}{2} – x$$ when $$0 < x < \frac{1}{2}$$, to $$\frac{1}{2}$$ when $$x = \frac{1}{2}$$, to $$\frac{3}{2} – x$$ when $$\frac{1}{2}< x < 1$$, 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 $$x = 0$$, $$x = \frac{1}{2}$$, and $$x = 1$$. Show also that the function may be represented by the formula $\tfrac{1}{2} – x – \tfrac{1}{2}[2x] – \tfrac{1}{2}[1 – 2x].$

18. Let $$\phi(x) = x$$ when $$x$$ is rational and $$\phi(x) = 1 – x$$ when $$x$$ is irrational. Show that $$\phi(x)$$ assumes every value between $$0$$ and $$1$$ once and once only as $$x$$ increases from $$0$$ to $$1$$, but is discontinuous for every value of $$x$$ except $$x = \frac{1}{2}$$.

19. As $$x$$ increases from $$-\frac{1}{2}\pi$$ to $$\frac{1}{2}\pi$$, $$y = \sin x$$ is continuous and steadily increases, in the stricter sense, from $$-1$$ to $$1$$. Deduce the existence of a function $$x = \arcsin y$$ which is a continuous and steadily increasing function of $$y$$ from $$y = -1$$ to $$y = 1$$.

20. Show that the numerically least value of $$\arctan y$$ is continuous for all values of $$y$$ and increases steadily from $$-\frac{1}{2}\pi$$ to $$\frac{1}{2}\pi$$ as $$y$$ varies through all real values.

21. Discuss, on the lines of §§ 108-109, the solution of the equations $y^{2} – y – x = 0,\quad y^{4} – y^{2} – x^{2} = 0,\quad y^{4} – y^{2} + x^{2} = 0$ in the neighbourhood of $$x = 0$$, $$y = 0$$.

22. If $$ax^{2} + 2bxy + cy^{2} + 2dx + 2ey = 0$$ and $$\Delta = 2bde – ae^{2} – cd^{2}$$, then one value of $$y$$ is given by $$y = \alpha x + \beta x^{2} + (\gamma + \epsilon_{x}) x^{3}$$, where $\alpha = -d/e,\quad \beta = \Delta/2e^{3},\quad \gamma = (cd – be) \Delta/2e^{5},$ and $${\epsilon_{x}}$$ is of the first order of smallness when $$x$$ is small.

[If $$y – \alpha x = \eta$$ then $-2e\eta = ax^{2} + 2bx(\eta + \alpha x) + c(\eta + \alpha x)^{2} = Ax^{2} + 2Bx \eta + C\eta^{2},$ say. It is evident that $$\eta$$ is of the second order of smallness, $$x\eta$$ of the third, and $$\eta^{2}$$ of the fourth; and $$-2e\eta = Ax^{2} – (AB/e) x^{3}$$, the error being of the fourth order.]

23. If $$x = ay + by^{2} + cy^{3}$$ then one value of $$y$$ is given by $y = \alpha x + \beta x^{2} + (\gamma + \epsilon_{x}) x^{3},$ where $$\alpha = 1/a$$, $$\beta = -b/a^{3}$$, $$\gamma = (2b^{2} – ac)/a^{5}$$, and $$\epsilon_{x}$$ is of the first order of smallness when $$x$$ is small.

24. If $$x = ay + by^{n}$$, where $$n$$ is an integer greater than unity, then one value of $$y$$ is given by $$y = \alpha x + \beta x^{n} + (\gamma + \epsilon_{x}) x^{2n-1}$$, where $$\alpha = 1/a$$, $$\beta = -b/a^{n+1}$$, $$\gamma = nb^{2}/a^{2n+1}$$, and $$\epsilon_{x}$$ is of the $$(n – 1)$$th order of smallness when $$x$$ is small.

25. Show that the least positive root of the equation $$xy = \sin x$$ is a continuous function of $$y$$ throughout the interval $${[0, 1]}$$, and decreases steadily from $$\pi$$ to $$0$$ as $$y$$ increases from $$0$$ to $$1$$. [The function is the inverse of $$(\sin x)/x$$: apply § 109.]

26. The least positive root of $$xy = \tan x$$ is a continuous function of $$y$$ throughout the interval $${[1, \infty)}$$, and increases steadily from $$0$$ to $$\frac{1}{2}\pi$$ as $$y$$ increases from $$1$$ towards $$\infty$$.