Let . Evaluate the limit
Let us consider the function . Then,
The following figure depicts the graph of the derivative of .
Consider the function . Order the numbers .
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,
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
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.