If $\underset{x\to 0}{\lim }{[1+x^2+\frac{f\left(x\right)}{x}]}^{\frac{1}{x^2}}$  is equal to $e^7$ , then   $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)}{{\mathrm{x}}^3}=..........$

$l=\underset{x\to 0}{\lim }{[1+\frac{x^3+f\left(x\right)}{x}]}^{1/x^2}$ if $A=\frac{x^3+f\left(x\right)}{x}$ ,
then  $l=\underset{x\to 0}{\lim }{\left\{{[1+A]}^{1/A}\right\}}^{A/x^2}=e^{\underset{x\to 0}{\lim }\frac{A}{x^2}}$
 $\underset{x\to 0}{\lim }\frac{A}{x^2}=\underset{x\to 0}{\lim }\frac{x^3+f\left(x\right)}{x^3}=\underset{x\to 0}{\lim }[1+\frac{f\left(x\right)}{x^3}]$
Now, $1+\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^3}=7,$ Given
 $\underset{x\to 0}{\lim }\frac{f\left(x\right)}{x^3}=6$