Limits of an Integration with Probability Density

Question

Let $X$ be random variable equipped with a density function $f_X$. I was bothering with the following equality.

$$ \lim_{h \to 0} \frac{1}{h} \int_x^{x+h}f_X(t)dt = f_X(x) $$

I don't see clearly why the above equality holds. This seems to me a bit like applying "fundamental theorem of calculus" but not quite sure how to massage it into the right looking. Any suggestion is appreciated. Thanks.

Answer 1

$$\frac1h\int_{x}^{x+h}f_X(x)\ dx=\frac{P(x\leq X\leq x+h)}{h}=$$ $$=\frac{F(x+h)-F(x)}h$$

and

$$\lim_{h\to 0}\frac{F(x+h)-F(x)}h=F_X'(x)=f_X(x)$$

if the density exists; and it does.