Can I always interpret a definite integral as a constant? For example, this problem
$$I = \lim_{x\to0^+} \frac{\int_a^bf(x) dx}{x^2}$$
Could I simply write: 
$$I = \left[{\int_a^bf(x) dx}\right] \lim_{x\to0^+} \frac{1}{x^2} = + \infty$$
 A: Yes. A definite integral (if it is finite) is just a way to write down a number so that anyone knows which number we are talking about. 
The same goes for $\pi$ etc. Just convenient ways to denote the same number everywhere.

For your case: it is not strictly prohibited, but kind of a bad style to use the same bound variable $x$ for both, the integral and the limit. Note that
$$\int_a^bf(x)\,\mathrm dx=\int_a^bf(y)\,\mathrm dy.$$
This means you should prefer writing your limit as
$$\lim_{x\to0^+}\frac{\int_a^bf(y)\,\mathrm dy}{x^2}$$
and the ambiguity is gone.

Finally it is important that even when you know your integral represents a number, you should always keep in mind that you may not know what number it is numerically. This is important because some rules of mathematics only a apply to certain numbers. For example
$$\lim \alpha x_n=\alpha\lim x_n$$
will hold (and only makes sense) if $\lim x_n$ converges in the first place. But if it does not converge neither does $\lim \alpha x_n$, right? Wrong! The value $\alpha$ could be zero which makes
$$\lim \alpha x_n=\lim 0=0,$$
hence convergent, while $\alpha \lim x_n$ might not be defined at all. This especially applies to your example:
$$\lim_{x\to0^+}\frac{\int_a^b f(y)\,\mathrm dy}{x^2} = \left[\int_a^bf(y)\,\mathrm dy\right]\cdot\lim_{x\to 0^+}\frac 1{x^2}$$
does not really make sense because $\lim_{x\to 0^+}1/x^2$ does not converge. Well, we can rescue the situation a bit by using the limit $+\infty$ and the rule $\alpha\cdot\infty=\infty$, right? Again, if the integral is zero or negative, this will give you wrong results.
Conclusion: do not use any rules which only work for certain number if you do not know the number because it is given implicitely (e.g. by an integral). First compute the number or its relevant properties (if possible).
