Hipi Zhdripi i Matematikës/1257

Stampa:Hipi Zhdripi i Matematikës kryefleta Stampa:Dygishta Mirëpo, pasi $\underset{x\to \mathrm{\infty }}{\lim }(-\frac{1}{x})=\underset{x\to \mathrm{\infty }}{\lim }\frac{1}{x}=0$, konkludojmë se $\underset{x\to \mathrm{\infty }}{\lim }\frac{\cos x}{x}=0$

Fig. 7.17.

Stampa:S h e m b u l l i Të njehsohet $\underset{x\to 0}{\lim }\frac{\sin x}{x}$

Stampa:Z g j i d h j e Në rrethin trigonometrik (fig. 7.17.) harkun $\stackrel{⌢}{AB}$ e shënojmë me $x$, atëherë:

Stampa:Dygishta 1 ° $S_{\mathrm{△}OAB}=\frac{OA\cdot CB}{2}=\frac{\sin x}{2}$

Stampa:Dygishta 2 ° $S_{\mathrm{△}(OAB)}=\frac{r\cdot \stackrel{⌢}{AB}}{2}=\frac{x}{2}$. dhe

Stampa:Dygishta 3° $S_{\mathrm{△}OAD}=\frac{OA\cdot AD}{2}=\frac{\mathrm{t}\mathrm{g}x}{2}$

Stampa:Dygishta Duke krahasuar këto madhësi marrim

$S_{\mathrm{△}OAB}<S_{\mathrm{△}(OAB)}<S_{\mathrm{△}OAD}$

ose

$\frac{\sin x}{2}<\frac{x}{2}<\frac{\mathrm{t}\mathrm{g}x}{2}$

nga del:

$\sin x<x<\mathrm{t}\mathrm{g}x\mathrm{o}\mathrm{s}\mathrm{e}1<\frac{x}{\sin x}<\frac{x}{\cos x}\mathrm{o}\mathrm{s}\mathrm{e}\cos x<\frac{\sin x}{x}<1$.

Pasi $\underset{x\to 0}{\lim }\cos x=1$, në bazë të teoremës 2.6.1., konkludojmë se

$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$.

Stampa:S h e m b u l l i Të njehsohet $\underset{x\to 0}{\lim }\frac{\sin 3x}{x}$

Stampa:Z g j i d h j e $\begin{array}{cc}\underset{x\to 0}{\lim }\frac{\sin 3x}{x}& =\left|\begin{array}{c}\mathrm{Z}\ddot{\mathrm{e}}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{.}3x=\alpha ;\\ \mathrm{k}\mathrm{u}\mathrm{r}x\to 0\Rightarrow \alpha \to 0\end{array}\right|=\underset{x\to 0}{\lim }3\cdot \frac{\sin \alpha }{\alpha }=\\ & =3\cdot \underset{x\to 0}{\lim }\frac{\sin \alpha }{\alpha }=3\end{array}$. Stampa:S h e m b u l l i Të njehsohet $\underset{x\to \mathrm{\infty }}{\lim }2x\cdot \sin \frac{1}{x}$.

Stampa:Z g j i d h j e $\begin{array}{cc}\underset{x\to \mathrm{\infty }}{\lim }2x\cdot \sin \frac{1}{x}& =\left|\begin{array}{c}\mathrm{Z}\ddot{\mathrm{e}}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{.}\frac{1}{x}=\alpha ;\\ \mathrm{k}\mathrm{u}\mathrm{r}x\to \mathrm{\infty }\Rightarrow \alpha \to 0\end{array}\right|=2\underset{x\to 0}{\lim }\frac{1}{\alpha }\cdot \sin \alpha =\\ & =2\underset{x\to 0}{\lim }\frac{\sin \alpha }{\alpha }=2.\end{array}$ Stampa:Hipi Zhdripi i Matematikës fundfleta