Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I want to prove that $\;\;\displaystyle \left|\frac{\sin x-x}{x^{2}}\right|\leq\frac{4(\pi/2-1)}{\pi^{2}}\;\;$ for all $x$ such that $x\in\left[0,\pi/2\right]$.

If you look at the graph of the expression on the left, it is clearly (appearing to be) monotonically increasing, so the maximum value of the left hand side is the output of the expression when $x=\pi/2$.

Would like a proof that does not involve calculus since this inequality is being used to prove $\displaystyle\lim_{x\rightarrow0}\frac{\sin x}{x}=1$.

share|improve this question
1  
Minor detail: you want $x$ in $(0,\pi/2]$. –  1015 Mar 5 '13 at 17:24
1  
But that inequality is quite stronger than $\lim\frac{\sin x}x=1$ (i.e. the limit follows immediately from the iniequality and just an arbitrary bound would be enough). So, what can we use about $\sin$? How is it defined? Purely geometrically? –  Hagen von Eitzen Mar 5 '13 at 17:24

2 Answers 2

You may already know that $\sin x < x< \tan x$ for $0<x<\frac\pi 2$. Then for such $x$ we have $$0< \frac{x-\sin x}{x^2}< \frac{\tan x-\sin x}{x^2}=\frac{(1-\cos x)\tan x}{x^2}=\frac{\sin^2x\tan x}{x^2(1+\cos x)}< \tan x.$$ This is not the bound you were asking for, but it is weaker only for big $x$, hence is more than enough to show $\lim_{x\to 0}\frac{\sin x}{x}=1$. (In fact, using $\cos x\to 1$, we obtain $\sin x=x+O(x^3)$)

share|improve this answer

Actually we can expect this function

$$f(x)=\dfrac{x -\sin x }{x^2}$$

as an increasing function.

$f'(x) = \dfrac{2\sin x-x\cos x-x}{x^3}$

we just have to prove

$g(x) = 2\sin x-x\cos x-x\ge 0$ , for $x\in[0,\pi/2]$

since $g(0) = 0$ , we just prove $g(x)$ is also increasing on $[0,\pi/2]$.

$g'(x) = \cos x+x\sin x -1$, with $g'(0) = 0$

and $g''(x) = x\cos x\ge 0$ on the interval $[0,\pi/2]$, thus $g'(x)$ is non-decreasing, thus $g'(x)\ge 0$, which means $g(x)$ is non-decreasing, thus $g(x)\ge 0.$

Therefore,

$f'(x)\ge 0$, $f$ is non-decreasing, thus $f(\dfrac{\pi}{2})$ will be the maximum.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.