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  • Evaluate the following limits : <img alt="\mathrm{\lim _{x \rightarrow 0} \frac{\sin x}{x} .}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7Bx%20%5Crightar

Evaluate the following limits :

\mathrm{\lim _{x \rightarrow 0} \frac{\sin x}{x} .}

Option: 1

1


Option: 2

0


Option: 3

\infty


Option: 4

None


Answers (1)

best_answer

Let the circular measure of each of the angles \mathrm{D O A} and \mathrm{D O B} be x where \mathrm{0<x<\frac{1}{2} \pi.}

Let the tangents at A and B meet \mathrm{ O D} produced in E and let the chord \mathrm{ A B} meet \mathrm{ O D} in \mathrm{ C, O} being the centre of the circle

\mathrm{B D A.} We shall assume now as an axiom the chord \mathrm{A C B<} the \mathrm{\operatorname{arc} A D B<A E+B E}. Dividing by \mathrm{O A}, we have

\mathrm{ \frac{\text { chord } A C B}{O A}<\frac{\operatorname{arc} A D B}{O A}<\frac{A E+B E}{O A} }

\mathrm{\text { or } \frac{2 A C}{O A}<\frac{2 \operatorname{arc} A D}{O A}<\frac{2 A E}{O A}\, \, \, \, \, \, \, \, \quad[\because A E=B E]}

\mathrm{\text { or } \sin x<x<\tan x}                      ....(1)

\mathrm{\text { or } \quad 1<x / \sin x<1 / \cos x \text {. }}

\mathrm{\text { Now let } x \rightarrow 0 \text {. Then } \frac{1}{\cos x} \rightarrow 1 \text {. Hence } \frac{x}{\sin x} \text { must }}

\mathrm{\text { also } \rightarrow 1}

\mathrm{\text { Therefore } \quad \lim _{x \rightarrow 0} \frac{\sin x}{x}=1 \text {. }}

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Divya Prakash Singh

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