4.1 The limit of a function

The concept of the derivative of a function is what most people think of as the beginning of calculus.However, before we can even define the derivative we must introduce a kind of generalization of the notion of continuity. That is, we must beginwith the definition of the limit of a function.

Let $f:S\to C$ be a function, where $S\subseteq C,$ and let $c$ be a limit point of $S$ that is not necessarily an element of $S.$ We say that $f$ has a limit L as z approaches c, and we write

$$\underset{z\to c}{lim}f\left(z\right)=L,$$

if for every $ϵ>0$ there exists a $\delta >0$ such that if $z\in S$ and $0<|z-c|<\delta ,$ then $\left|f\right(z)-L|<ϵ.$

If the domain $S$ is unbounded, we say that f has a limit L as z approaches $\infty ,$ and we write

$$L=\underset{z\to \infty }{lim}f\left(z\right),$$

if for every $ϵ>0$ there exists a positive number $B$ such that if $z\in S$ and $\left|z\right|\ge B,$ then $\left|f\right(z)-L|<ϵ.$

Analogously, if $S\subseteq R,$ we say ${lim}_{x\to \infty }f\left(x\right)=L$ if for every $ϵ>0$ there exists a real number $B$ such that if $x\in S$ and $x\ge B,$ then $\left|f\right(x)-L|<ϵ.$ And we say that ${lim}_{x\to -\infty }f\left(x\right)=L$ if for every $ϵ>0$ there exists a real number $B$ such that if $x\in S$ and $x\le B,$ then $\left|f\right(x)-L|<ϵ.$

Finally, for $f:(a,b)\to C$ a function of a real variable, and for $c\in [a,b],$ we define the one-sided (left and right) limits of $f$ at $c.$ We say that $f$ has a left hand limit of $L$ at $c,$ and we write $L={lim}_{x\to c-0}f\left(x\right),$ if for every $ϵ>0$ there exists a $\delta >0$ such that if $x\in (a,b)$ and $0<c-x<\delta$ then $\left|f\right(x)-L|<ϵ.$ We say that $f$ has a right hand limit of $L$ at $c,$ and write $L={lim}_{x\to c+0}f\left(x\right),$ if for every $ϵ>0$ there exists a $\delta >0$ such that if $x\in S$ and $0<x-c<\delta$ then $\left|f\right(x)-L|<ϵ.$

The first few results about limits of functions are not surprising. The analogy between functions having limits and functions being continuous is very close,so that for every elementary result about continuous functions there will be a companion result about limits of functions.

Let $c$ be a complex number. Let $f:S\to C$ and $g:S\to C$ be functions. Assume that both $f$ and $g$ have limits as $x$ approaches $c.$ Then:

1. There exists a $\delta >0$ and a positive number $M$ such that if $z\in S$ and $0<|z-c|<\delta$ then $\left|f\right(z\left)\right|<M.$ That is, if $f$ has a limit as $z$ approaches $c,$ then $f$ is bounded near $c.$
2. $$\underset{z\to c}{lim}(f\left(z\right)+g\left(z\right))=\underset{z\to c}{lim}f\left(z\right)+\underset{z\to c}{lim}g\left(z\right).$$
3. $$\underset{z\to c}{lim}\left(f\left(z\right)g\left(z\right)\right)=\underset{z\to c}{lim}f\left(z\right)\underset{z\to c}{lim}g\left(z\right).$$
4. If ${lim}_{z\to c}g\left(z\right)\ne 0,$ then
   $$\underset{z\to c}{lim}\frac{f\left(z\right)}{g\left(z\right)}=\frac{{lim}_{z\to c}f\left(z\right)}{{lim}_{z\to c}g\left(z\right)},$$
5. If $u$ and $v$ are the real and imaginary parts of a complex-valued function $f,$ then $u$ and $v$ have limits as $z$ approaches $c$ if and only if $f$ has a limit as $z$ approaches $c.$ And,
   $$\underset{z\to c}{lim}f\left(z\right)=\underset{z\to c}{lim}u\left(z\right)+i\underset{z\to c}{lim}v\left(z\right).$$

REMARK When we use the word “ interior” in connection with a set $S,$ it is obviously important to understand the context; i.e., is $S$ being thought of as a set of real numbers or as a set of complex numbers.A point $c$ is in the interior of a set $S$ of complex numbers if the entire disk $B_{ϵ}\left(c\right)$ of radius $ϵ$ around $c$ is contained in $S.$ While, a point $c$ belongs to the interior of a set $S$ of real numbers if the entire interval $(c-ϵ,c+ϵ)$ is contained in $S.$ Hence, in the following definition, we will be careful to distinguish between the cases that $f$ is a function of a real variable or is a function of a complex variable.