The quadratic equation whose roots are $l$ and $m$, where $$ \begin{aligned} & l=\lim _{ heta ightarrow 0}\left(\frac{3 \sin heta-4 \sin ^2 heta}{ heta} ight), \ & m=\lim _{ heta ightarrow 0} \frac{2 an heta}{ heta\left(1- an ^2 heta ight)}, ext { is } \end{aligned} $$

Question

The quadratic equation whose roots are $l$ and $m$, where $$ \begin{aligned} & l=\lim _{	heta ightarrow 0}\left(\frac{3 \sin 	heta-4 \sin ^2 	heta}{	heta}ight), \ & m=\lim _{	heta ightarrow 0} \frac{2 	an 	heta}{	heta\left(1-	an ^2 	hetaight)}, 	ext { is } \end{aligned} $$

Aicharya · Accepted Answer

We have, $ l=\lim _{	heta ightarrow 0}\left[\frac{3 \sin 	heta-4 \sin ^2 	heta}{	heta}ight] $ Using $L$-hospital's rule $ =\lim _{	heta ightarrow 0} \frac{3 \cos 	heta-8 \sin 	heta \cos 	heta}{1}=3 $ and $ \begin{aligned} m & =\lim _{	heta ightarrow 0} \frac{2 	an 	heta}{	heta\left(1-	an ^2 	hetaight)} \ & =\lim _{	heta ightarrow 0} \frac{	an 2 	heta}{	heta} \ & =\lim _{	heta ightarrow 0} \frac{	an 2 	heta}{2 	heta} \cdot 2=2 \end{aligned} $ The required quadratic equation is $ \begin{aligned} & x^2-(l+m) x+l m=0 \ & \Rightarrow \quad x^2-(3+2) x+3 \cdot 2=0 \ & x^2-5 x+6=0 \ & \end{aligned} $

The quadratic equation whose roots are $l$ and $m$, where $$ \begin{aligned} & l=\lim _{\theta \rightarrow…

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