Calculus Problem Solutions with MATLAB®

3 Limits

Limit problems are the mathematical foundation of calculus. It is even noted in [24] that “Limit is the study of calculus”.

The research in calculus originally started from the limit problems. For instance, the Greek astronomer and mathematician Eudoxus of Cnidus (c408bce–c355bce) proposed the method of exhaustion, while in ancient China, the philosopher Zhuang Zhou (also known as Chuang Tzu, c369bce–286bce) stated that “For a foot-long stick, if you take half from the remaining part each day, you will never exhaust it in a million years”. The mathematical problem of the above can be expressed as follows:

$\begin{array}{c}\mathit{R}=1-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2^2}}-{\displaystyle \frac{1}{2^3}}-{\displaystyle \frac{1}{2^4}}-\cdots .\end{array}$

Mathematicians and philosophers, of course, can imagine “never exhaust in million years”, however, scientists ...