The quadratic equation whose roots are $m$ and $n$, where $m=\lim _{x ightarrow 0} \frac{x \log (1+2 x)}{x an x}$ and $n=\lim _{x ightarrow 0} \frac{\log x+\log \left(\frac{(1+x)}{x} ight)}{x}$, is

Question

The quadratic equation whose roots are $m$ and $n$, where $m=\lim _{x ightarrow 0} \frac{x \log (1+2 x)}{x 	an x}$ and $n=\lim _{x ightarrow 0} \frac{\log x+\log \left(\frac{(1+x)}{x}ight)}{x}$, is

Aicharya · Accepted Answer

$\begin{aligned} & m=\lim _{x ightarrow 0} \frac{x \log (1+2 x)}{x \cdot 	an x}=\lim _{x ightarrow 0} \frac{\frac{\log (1+2 x)}{2 x} \cdot 2 x}{\frac{	an x}{x} \cdot x}=2 \ & n=\lim _{x ightarrow 0} \frac{\log x+\log \left(\frac{1+x}{x}ight)}{x}=\lim _{x ightarrow 0} \frac{\log \left(x 	imes \frac{1+x}{x}ight)}{x} \ & =\lim _{x ightarrow 0} \frac{\log (1+x)}{x}=1\end{aligned}$ now, required quadratic equation is $x^2-(m+n) x+m n=0$ $\Rightarrow x^2-3 x+2=0$

The quadratic equation whose roots are $m$ and $n$, where $m=\lim _{x \rightarrow 0} \frac{x \log (1+2 x)}{x…

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