MISCELLANEOUS EXAMPLES ON CHAPTER V

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

 

2. If $P(x)=a{x}^n+b{x}^{n-1}+\cdots +k$, 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 $$(a{x}^n+b{x}^{n-1}+\cdots +k)/(A{x}^n+B{x}^{n-1}+\cdots +K)=\alpha +(\beta /x)(1+{ϵ}_x),$$ where $\alpha =a/A$, $\beta =(bA–aB)/A^2$, and ${ϵ}_x$ is of the first order of smallness when $x$ is large. Indicate any exceptional cases.

 

4. Express $$(a{x}^2+bx+c)/(A{x}^2+Bx+C)$$ in the form $$\alpha +(\beta /x)+(\gamma /x^2)(1+{ϵ}_x),$$ where ${ϵ}_x$ is of the first order of smallness when $x$ is large.

 

5. Show that $$\underset{x\to \mathrm{\infty }}{lim}\sqrt{x}\{\sqrt{x+a}–\sqrt{x}\}=\frac{1}{2}a.$$

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

 

6. Show that $\sqrt{x+a}=\sqrt{x}+\frac{1}{2}(a/\sqrt{x})(1+{ϵ}_x)$, where ${ϵ}_x$ 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 $x\to \mathrm{\infty }$; and prove that $limx\{\sqrt{a{x}^2+2bx+c}–\alpha x–\beta \}=(ac–b^2)/2a$.

 

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

 

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

 

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

 

11. Prove that $\varphi (x)=x\sin (\sin x)–{\sin }^{2}x$ is of the sixth order of smallness when $x$ is small; and find the limit of $\varphi (x)/x^6$ 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; $\mathrm{\Delta }$ is the area of the triangle formed by the chord of the arc and the two tangents at the extremities, and ${\mathrm{\Delta }}^{\prime }$ the area of that formed by the three tangents. Show that $\mathrm{\Delta }/{\mathrm{\Delta }}^{\prime }\to 4$ as the length of the arc tends to zero.

 

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

 

15. If $\varphi (x)=1/q$ when $x=p/q$, and $\varphi (x)=0$ when $x$ is irrational, then $\varphi (x)$ 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 $$1–x+[x]–[1–x],1–x–\underset{n\to \mathrm{\infty }}{lim}({\cos }^{2n+1}\pi x).$$

 

17. Show that the function $\varphi (x)$ 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 $$\frac{1}{2}–x–\frac{1}{2}[2x]–\frac{1}{2}[1–2x].$$

 

18. Let $\varphi (x)=x$ when $x$ is rational and $\varphi (x)=1–x$ when $x$ is irrational. Show that $\varphi (x)$ 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 $$y^2–y–x=0,y^4–y^2–x^2=0,y^4–y^2+x^2=0$$ in the neighbourhood of $x=0$, $y=0$.

 

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

[If $y–\alpha x=\eta$ then $$-2e\eta =a{x}^2+2bx(\eta +\alpha x)+c(\eta +\alpha x{)}^2=A{x}^2+2Bx\eta +C{\eta }^2,$$ 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 $-2e\eta =A{x}^2–(AB/e)x^3$, the error being of the fourth order.]

 

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

 

24. If $x=ay+b{y}^n$, where $n$ is an integer greater than unity, then one value of $y$ is given by $y=\alpha x+\beta x^n+(\gamma +{ϵ}_x)x^{2n-1}$, where $\alpha =1/a$, $\beta =-b/a^{n+1}$, $\gamma =n{b}^2/a^{2n+1}$, and ${ϵ}_x$ 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 $[1,\mathrm{\infty })$, and increases steadily from $0$ to $\frac{1}{2}\pi$ as $y$ increases from $1$ towards $\mathrm{\infty }$.

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$\leftarrow$ 108-109. Implicit and inverse functions

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