Question

If $\underset{x\to \infty }{lim}f\left(x\right)$ exists and is finite and nonzero and if $\underset{x\to \infty }{lim}\left\{f\right(x)+\frac{3f\left(x\right)-1}{f^2\left(x\right)}\}=3$, then the value of $\underset{x\to \infty }{lim}f\left(x\right)$ is ............

• A. $2$
• B. $1$
• C. $-1$
• D. $3$

Answer 1

Answer: (B)

$1$

$\underset{x\to \infty }{lim}\left(f\right(x)+\frac{3f\left(x\right)-1}{f^2\left(x\right)})=3$
Let $\underset{x\to \infty }{lim}f\left(x\right)=y$
or $\left(\underset{x\to \infty }{lim}f\left(x\right)+\frac{3{lim}_{x\to \infty }f\left(x\right)-1}{{\left(\underset{x\to \infty }{lim}f\left(x\right)\right)}^2}\right)=3$
or $(y+\frac{3y-1}{y^2})=3$
or $y^3-3y^2+3y-1=0$
or $(y-1{)}^3=0$
or $y=1$
Hence, option 'A' is correct.