$\lim\limits_{n\rightarrow\infty} \left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}\right)$
. Hint: Graph sketching may help.

 Sketch
$y=\frac{x}{x^2a^2}$
for various values of$a$
.  Integrate
$y=x\ln x$
.  Evaluate
$\lim\limits_{x\to\infty} \big(\ln x\ln(2x1)\big).$
 Sketch

What does the graph of
$\frac{1}{x}$
look like? 
On the above graph, try representing each term of the sum as a (very basic) shape with an area equal to the term’s value.

Which 2 continuous functions can you use as upper and lower bounds for the terms of the sum, having this new representation? Hint: one of them you already used.

Consider the graph below. How do you relate the areas underneath the two functions to the sum?

Squeeze theorem?

Each number
$\frac{1}{k}$
is equal to the area of the rectangle extending to the right of the number, having height$\frac{1}{k}$
and width 1. By sketching, notice there are two continuous functions bounding these rectangles, one above, i.e.$\frac{1}{x1}$
, and one below, i.e.$\frac{1}{x}$
, whose integrals will also bound the initial sum under the limit.Concretely,
$$ \int_{n+1}^{2n+1} \frac{dx}{x} < \sum_{k=n+1}^{2n}\frac{1}{k} < \int_{n+1}^{2n+1}\frac{dx}{x1}. $$
At$n\rightarrow\infty$
, both integrals become$\ln2,$
and by the squeeze theorem, so must the sum.