Let $f(x)0$ be a continuous function on the interval $a,b,$ and let $a,b$ be subdivided into $n$ intervals of equal width $x.$ Let $x_i=a+ix$ for $i=0,$dots,$n.$ Let $R_n$ and $L_n$ be the right and left endpoint approximations, respectively, of the area under $f(x)$ on the interval $a,b,$ and let $A$ be the exact area under the curve.

Below are four statements. Select EVERY true statement $($there may be more than one$).$

$L_n={\displaystyle \underset{i=0}{\overset{n-1}{}}}f(x_i)x$

$R_n={\displaystyle \underset{i=1}{\overset{n}{}}}f(x_i)x$

$\underset{n+}{lim}{R}_n=A$

$\underset{?}{lim}{R}_n=\underset{n}{lim}{t}_n$