Definite integral is the limit of Riemann sum.
$\lim\limits_{x \to a} f(x)=L$ is defined as:
For every $\varepsilon>0$, we can find a $\delta>0$ such that $\left| {f\left( x \right) - L} \right| < \varepsilon \hspace{0.5in}{\mbox{whenever}}\hspace{0.5in}0 < \left| {x - a} \right| < \delta$
In an analogous way (or in a different way), how can we define the limit of a sum?
The formal definition of an integral is that $\int_a^b f(x)dx=I$ if for every $\epsilon>0$ we can find $\delta>0$ such that for any partition $a=x_0<x_1<...<x_n=b$ of $[a,b]$ which satisfies $\max (x_i-x_{i-1})<\delta$ and for any choice of points $\{t_i\}_{i=1}^n, t_i\in [x_{i-1},x_i]$ we have that the distance of the corresponding Riemann sum from $I$ is at most $\epsilon$. So it is more than just a limit, it must not depend on partition and choice of points.