# Math Help - limit help

1. ## limit help

Let f be a continuous function defined on [a,b]. Find the limit as n approaches infinity of the integral of (nf) from a to (a+1)/n.

2. $(a+1)/n$ is not always in $[a,b]$, don't you mean $a+(1/n)$ (for sufficiently large $n$)

Edit: Assuming you mean $a+(1/n)$ take $F(x)= \int_{a} ^{x} f(t)dt$ then $F$ is differentiable in $(a,b)$ and it's derivative is continous on $[a,b]$ ie. F has left derivative at $b$ and and right derivative at $a$. $\lim_{n \rightarrow \infty } n\int_{a} ^{a+(1/n)} f = \lim_{n \rightarrow \infty } nF(a+(1/n)) = \lim_{n \rightarrow \infty } \frac{F(a+(1/n))-F(a)}{1/n} =F'(a)=f(a)$