Question

If $\underset{x\to \infty }{lim}[1+x+\frac{f\left(x\right)}{x}{]}^{\frac{1}{x}}=e^3$ then the value of the function $f\left(x\right)$ may be

• A. $\frac{x^2}{2}$
• B. $x^2$
• C. $2x^2$
• D. $3x^2$

Answer 1

The correct option is C $2x^2$

$\underset{x\to \infty }{lim}{[1+x+\frac{f\left(x\right)}{x}]}^{\frac{1}{x}}=e^3$
or $\underset{x\to \infty }{lim}{e}^{{lim}_{x\to \infty }{[1+x+\frac{f\left(x\right)}{x}1]}^{\frac{1}{x}}}=e^3$
or $\underset{x\to \infty }{lim}{e}^{{lim}_{x\to \infty }[1+\frac{f\left(x\right)}{x^2}]}=e^3$
or $\underset{x\to \infty }{lim}\frac{f\left(x\right)}{x^2}=2$

Then $f\left(x\right)=2x^2$