Knowledge of Limits in mathematics

The knowledge of limits is fundamental to mathematics and plays a crucial role in various branches such as calculus. Understanding limits helps mathematicians and scientists analyse the behaviour of functions, study continuity, and explore the concept of infinity. Here are some key points about the value of knowing limits:

Foundations of calculus:

Limits are fundamental to the development of calculus, allowing the study of derivatives and integrals.
The concept of limit is essential for understanding instantaneous rates of change, slopes of curves, and areas under curves.

Analysis of functions: Limits allow mathematicians to study the behaviour of functions as the input approaches a particular value, providing insight into the overall behaviour of the function. They are used to determine whether a function is continuous at a given point.

Infinite processes: Limits are used to study infinite processes, such as infinite series and sequences.
Calculus uses limits to deal with situations where quantities approach infinity or become infinitesimally small.

Regarding the applicability of infinitesimal behaviour to real life, it’s important to note that while the concept of limits and infinitesimals is fundamental to mathematics, the idea of true infinitesimals (quantities that are infinitely small) is not always directly applicable to the physical world. In reality, there is a limit to how small a quantity can be measured due to the limitations of physical instruments.

However, the concept of approaching a limit or infinitesimal is still valuable in modelling and understanding real world phenomena. In many scientific and engineering applications, calculus and limits are used to approximate and analyse physical processes effectively.

As for intuition in mathematics, it is indeed a valid way of knowing. Mathematical intuition is the ability to make educated guesses, recognise patterns and develop a ‘feel’ for mathematical concepts. While rigorous proof and logical reasoning are essential in mathematics, intuition often guides mathematicians in formulating conjectures, making discoveries and finding creative solutions.

Intuition can be developed through experience and exposure to mathematical concepts over time. It complements formal reasoning and can lead to insights that drive further exploration and development of mathematical theories. However, it’s important to balance intuition with rigorous proof and verification to ensure the validity of mathematical statements and conclusions.