Question

If $\displaystyle\lim _{x \rightarrow \infty} f(x)$ exists and is finite and non-zeros and if $\displaystyle\lim _{x \rightarrow \infty}\left(f(x)+\frac{3 f(x)-1}{f^{2}(x)}\right)=3$, then the value of $\displaystyle\lim _{x \rightarrow \infty} f(x)$ is

Answer 1

$\displaystyle\lim _{x \rightarrow \infty}\left(f(x)+\frac{3 f(x)-1}{f^{2}(x)}\right)=3$
$\Rightarrow \left(\displaystyle\lim _{x \rightarrow \infty} f(x)+\frac{3 \displaystyle\lim _{x \rightarrow \infty} f(x)-1}{\left(\displaystyle\lim _{x \rightarrow \infty} f(x)\right)^{2}}\right)=3$
$\Rightarrow \left(y+\frac{3 y-1}{y^{2}}\right)=3$
$\Rightarrow y^{3}-3 y^{2}+3 y-1=0$
$\Rightarrow (y-1)^{3}=0$
$\Rightarrow y=1$