Calculate a limit with an integral within proving that it's possible to use the L'Hôpital rule Let $f: [1, +\infty) \rightarrow R\;$ be a continuous function, bounded, and such that $f(x) \ge1 \;\;\;\forall\;x\ge1$. Calculate reasonably the following limit, proving that it is possible to use L'Hôpital Rule:
$$\lim_{x\to +\infty} \frac{1}{x} \int_{1}^{x^2} \frac{f(t)}{t}dt$$
I have been trying to prove we can use L'Hôpiatl rule by giving examples of functions which meet those conditions, such as the aditive polynomical, irrational (where the degree of the numerator is higher than the one of the denominator) and exponential functions, but then I'm stuck and I don't know how to continue.
Thank you! 

For conditions under which L'Hospital is applicable refer to Paramanand Singh's answer.
L'Hospital:
Numerator: FTC and Chain Rule.
$\lim_{x \rightarrow \infty} \dfrac{\displaystyle{\int_{1}^{x^2}}(f(t)/t)dt}{x}=$
$\lim_{x \rightarrow \infty} \dfrac{(f(x^2)/x^2)(2x)}{1}=$
$\lim_{x \rightarrow \infty} 2\dfrac{f(x^2)}{x}.$
