## The limit is zero

Let be an integrable and uniformly continuous function. Prove that.

**Solution**

*This exercise is quite known. It first made its appearance as a Berkley exam question.*

Suppose that the limit is not zero. Then let us pick a sequence and an such that foreach to both hold and . Since is uniformly continuous there exists such that

However in the interval it holds that

foreach positive. The last is a consequence of . And of course this contradicts the fact that the function is integrable. Hence the conclusion.

**Note: **We can’t drop the condition of uniform continuity. If so, then the function is a counterexample. Indeed it is continuous, integrable but the limit at infinity is not .

## An integral inequality

Let be a continuous function such that

(1)

Prove that .

**Solution**

and thus the conclusion.

## On a determinant

Let be a prime number and let be a primitive p-th root of unity. Define:

Evaluate the rational number .

**Solution**

since it is known that for any -th root of unity rather than . Thus:

Thus is there is a at the upper left corner and ‘s along the anti diagonal in the lower right block. Thus:

## A limit of a sum

Evaluate the limit

**Solution**

The fastest way is by making use of probabilities. Let us consider independent Poisson distributions with parameter . Then is Poisson with parameter . From the central limit theorem converges in distribution to the standard normal distribution. In particular, if follows a standard normal distribution then

and the conclusion follows.

## An infinite product with Fibonacci

Compute the product

where is the -th Fibonacci sequence term.

**Solution**

where is the golden ratio and its value is .