Is limit about functions?

Question

When talking about limit subject, all the time we point functions. But in derivative definition we use $\frac{\Delta Y}{\Delta X}$. Is $\frac{\Delta Y}{\Delta X}$ a kind of function? I have this question about integral formula as well.

Answer 1

Yes, $\Delta Y/\Delta X$ is a kind of limit. So is the integral. But there are some nuances that can make this confusing.

$\Delta Y/\Delta X$ is an older, traditional notation. This can be helpful but also confusing. That's why many modern treatments use something like $$\frac{f(x+h)-f(x)}{h}$$ instead. Here, $f$ is the function $X\mapsto Y$ (so $Y=f(X)$), $h=\Delta X$, and $\Delta Y=f(x+h)-f(x)$. The newer notation makes explicit the function $f$. It also gives us the expression $(f(x+h)-f(x))/h$ to parse.

Strictly speaking, $(f(x+h)-f(x))/h$ is a function of two variables: $x$ and $h$. We could write it $D(x,h)=(f(x+h)-f(x))/h$. It is undefined when $h=0$. If we pick a particular value for $x$ and let $h$ vary, then it becomes a function of just one variable, $h$. People often use the term parameter for something like $x$: a variable whose value we fix, to focus attention on the remaining variables.

Since $(f(x+h)-f(x))/h$ is a function of $h$, but undefined at $h=0$, we can now try to take the limit $\lim_{h\to 0}(f(x+h)-f(x))/h$. If the limit exists, then $f$ is differentiable for that value of $x$. Here, $h$ is a so-called dummy or bound variable in the limit expression: the limit, if it exists, does not depend on $h$, but only the function $f$ and the fixed value $x$. "Unfixing" the value $x$, we then have a function of $x$, namely $f'(x)$.

In a sense, $f$ itself is a variable in the difference quotient. You can't evaluate $(f(x+h)-f(x))/h$ until you know what your function $f$ is. Of course, it's a different kind of variable: not a real number, but a function from real numbers to real numbers. So we can even consider the function $f\mapsto f'$, the "derivative function", whose domain is the collection of all differentiable functions.

With that in mind, look at the integral $$\int_a^b f(x) dx$$ This is a function of $a$, $b$, and $f$. The variable $x$ is again a bound or dummy variable: the integral does not depend on it. In more advanced treatments, you'll often see just $\int f$.

If we fix $a$ and $f$ and just let $b$ vary, we now have a function of one variable, and the famous fundamental theorem of calculus (in one form) says that the derivative of this function is $f$. We can also study $f\mapsto \int f$ as a "function of functions".

The definition of the integral $\int_a^b f(x) dx$ as a limit is considerably more complicated than the definition of the derivative as a limit. The simplest version is the Riemann integral with evenly spaced partitions, and sample points at the right: $$\int_a^b f(x)dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x$$ where $\Delta x=\frac{b-a}{n}$ and $x_i = a+i\Delta x$. Inside the summation symbol we have an expression which is a function of $f$, $x_i$ and $\Delta x$. But the definitions of these in terms of $a$, $b$, $i$, and $n$, makes this a function of $f$, $a$, $b$, $i$, and $n$. Summing on $i$ turns $i$ into a dummy variable. Taking the limit as $n\to\infty$ leaves us with a function of $f$, $a$, and $b$, the definite integral.