Stolz's Theorem

Theorem.

Assume that $(x_n)_{n=1}^{\infty}$ and $(y_n)_{n=1}^{\infty}$ are two sequences of real numbers such that $y_n$ is strictly increasing and $\lim_{n\to\infty}y_n=+\infty$. Then,

1. (a) the following three inequalities hold:
   \[
    \liminf \frac{x_{n+1}-x_n}{y_{n+1}-y_n}\leq \liminf\frac{x_n}{y_n}\leq \limsup\frac{x_n}{y_n}\leq \limsup \frac{x_{n+1}-x_n}{y_{n+1}-y_n}.
   \]
2. (b) Assume that $\displaystyle\lim_{n\to\infty}\frac{x_{n+1}-x_n}{y_{n+1}-y_n}=L$. Then the limit $\displaystyle \lim_{n\to\infty}\frac{x_n}{y_n}$ exists and is equal to $L$. 

Sieve formula

Exercise.

In a university 60% of students learn analysis, 50% learn algebra and 50% learn set theory. We know that 30% learn analysis and algebra, 20% learn algebra and set theory, 40% learn analysis and set theory. 10% of students learn analysis, algebra and set theory.
What percentage of students
(a) do not learn any mentioned topic?
(b) study exactly two topics?
(c) study at least two topics?

Maple

We solve a lot of exercises (from easy to difficult) using the computer algebra system Maple.

Snell's Law

A nice application of the calculation of extremal value.

Calculation of $\displaystyle{\lim_{x_0}}f(x)^{g(x)}$

Theorem.

Let $x_0$ be a finite real number, $\infty$ or $-\infty$.
Assume that

1. (a)  $\displaystyle{\lim_{x_0}}\,f(x)=1$;
2. (b)  $\displaystyle{\lim_{x_0}}\,g(x)(f(x)-1)=b.$

Then we have \[ \lim_{x_0}f(x)^{g(x)}=e^b. \]