## Rational limit

Let . Evaluate the limit

Solution

Let us consider the function . Then,

Hence,

## Ordering values

The following figure depicts the graph of the derivative of .

Consider the function . Order the numbers .

Solution

We are working on the following figure.

The area included by , the axis and the lines , is less than . Hence,

On the other hand . Hence,

Hence,

## Double inequality

Prove the following double inequality, where the sum and product are cyclic over the angles of a triangle

Given a cyclic quadrilateral inscribed in a semicircle of diameter as shown at the figure with and sides of lengths and

show that:

Solution

We recall Ptolemy’s theorem.

If a quadrilateral is inscribable in a circle then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides.

Applying Pythagoras’  Theorem to both and along with Ptolemy’s Theorem we get the result.

## Limit with harmonics

Let denote the -th harmonic number. Evaluate the limit

Solution

It follows from Cesaro and the definition of the Euler Mascheroni constant that the RHS is equal to

The first two terms tend to . All that is left to evaluate the limit of which equals from here.