A limit involving floor functions

Question

Looking for help on calculating the following limit : $$\lim_{a\to \infty}\,\frac{\lfloor{\frac{{1+\sqrt{a}}}{2}}\rfloor+\lfloor{\frac{a}{2}}\rfloor}{a},$$ if it exists. It appears to approach $\frac{1}{2}$, but I am unsure how to show this.

Answer 1

Call $f(a)$ your sum. Then $$ \frac{\frac{1+\sqrt{a}}{2}+\frac{a}{2}-2}{a} \le f(a) \le \frac{\frac{1+\sqrt{a}}{2}+\frac{a}{2}}{a}. $$ Since lhs and rhs converge both to $1/2$, then $f(a)\to 1/2$.