In calculus, the L’Hôpital’s rule states that the limit of a quotient of functions is equal to the limit of its derivatives, under certain conditions. A statement and a proof of the L’Hôpital’s rule for 0/0 form Result (L’Hôpital’s rule). Let \(f\) and \(g\) be real-valued differentiable functions over an open interval \(I\) except possibly…

# Tag: functions

## The average rate of change

The average rate of change of a function can be calculated by dividing the difference between the dependent variable and the independent variable. In this post, we discuss how to compute the average rate of change of functions in different spaces. The average rate of change in single variable calculus Let \(y = f(x)\) be…

## Limit of a function and laws

In this post, we discuss the concept of the limit of a function and review the limit laws. The limit of a function is to investigate the behavior of a function near a point. This is even though the value of the function at that point is not definable or unknown. What we say about…

## Single-variable functions

School mathematics teaches simple functions with one variable. Higher mathematics teaches more complex functions. In higher mathematics, we also learn some theoretical aspects of single-variable functions such as limits, continuity, differentiability, and integrability. In this post, we review some definitions and terminology related to single-variable functions. Definition of single-variable functions A function is a special…

## Orthogonal linear maps

Orthogonal linear maps are generalizations of orthogonal matrices. By definition, a linear map \(f\) over an inner product real vector space \(V\) is an orthogonal linear map (function, or transformation) if it preserves the inner product, i.e. for all vectors \(u\) and \(v\) in \(V\), we have $$ f(u) \cdot f(v) = u \cdot v.$$…

## 1-to-1 and onto linear maps

Linear maps are essential in linear algebra, and 1-to-1 and onto linear maps are especially important. In this post, we investigate 1-to-1 and onto linear maps and discuss them with suitable examples. 1-to-1 linear maps A function is, by definition, 1-to-1 (also called one-to-one or injective) if \(f(x) = f(y)\) implies \(x=y\), for all \(x\)…

## Two-variable functions

Two-variable functions play an essential role in multi-variable calculus. They are also essential in different branches of abstract algebra including linear algebra. In this post, first, we discuss two-variable functions from different points of view. Then, we proceed to introduce their role in abstract algebra. A two-variable function is a function whose input is an…

## Linear functions and examples

Since, in linear algebra, linear combination of vectors play an essential role, a function that preserves linearity must be of special importance also. Such functions are called linear functions. Note that in single variable calculus, by a linear function, it is meant a function \( f\) from \( \mathbb{R} \) into \( \mathbb{R} \) defined…

## Convergent real sequences

Convergent real sequences play a crucial role in mathematics for data science. In this post, we will first define real sequences and their convergence. We, then, list some basic properties of convergent real sequences. Finally, we give some examples to clarify the crucial points related to the convergence of real sequences. Convergent real sequences Definition…