Limits

We introduce the concept of limits with the example of tangent lines and, consequently, rates of change. Consider a function $y=f(x)$, as shown in Fig. 2.1. Two points are indicated, $\mathrm{P}$ and $\mathrm{Q}$ with $x$-coordinates at $x$ and $x+\Delta x$, respectively. Suppose we wanted to find the slope of the line tangent, shown in green in Fig. 2.1, to $y=f(x)$ at $x$ (this is the meaning of the derivative). To look for the slope at $P$, we can start with the secant line going through points $\mathrm{P}$ and $\mathrm{Q}$, shown in blue in Fig. 2.1. The secant line has slope,

We can perhaps see that we can approximate the tangent line at $\mathrm{P}$ by moving the point $\mathrm{Q}$ to the left, towards $P$ thus decreasing the distance $\Delta x$. In the limit $\Delta x \rightarrow 0$, we have

which gives the slope of the tangent line at P. In Eq. (2.2), we take the limit of a quantity as $\Delta x$ gets closer to 0 .

More generally, the limit of $f(x)$ is a finite number, $L$ as $x$ approaches some number $a$

provided that $f(x)$ approaches $L$ for all $x$ sufficiently close to $a$ from both sides; i.e. from the left (denoted as $x \rightarrow a^{-}$) and from the right (denoted as $x \rightarrow a^{+}$). An alternative notation for denoting limits is $f(x) \rightarrow L$ as $x \rightarrow a$.

Or, more formally, a limit is defined as:

Definition 1.1 Let $x=a$ be a point in the domain of $g(x)$. Then,

if for every number $\epsilon>0$, there exists some number, $\delta>0$ such that,

Notice that we do not say that $x$ is equal to $a$; we say $x$ approaches a (either from the left or from the right). At $x=a$, we do not get the limit but, instead, we get the value $g(a)$. Of course we can see from the graph that $g(a)=L$ so why do we really need to worry about the numbers $\epsilon$ and $\delta$ ? At this point let us define continuity: we can only plug the limit into the function if the function is continuous (see Definition 1.2).

A function $g(x)$ is said to be continuous at $x=a$ if,

Further, the function is said to be continuous on an interval, $\mathcal{I}$ if it is continuous at every point on the interval.

Definitions

When studying limits, we are concerned with the function around the point $x=a$ and not with the actual point $x=a$. It is possible that the function is not defined at a particular point but the limit as the point is approached, is finite. For instance, consider the function

Figure 2.1: The graph of a function $y=f(x)$ showing two points $\mathrm{P}=(x, f(x))$ and $\mathrm{Q}=(x+\Delta x, f(x+\Delta x))$. The line in green is the tangent line at point $\mathrm{P}$. The line in blue is a secant line between the points $\mathrm{P}$ and Q.

where $f(1)$ is undefined; however a graph of the function [see Fig. 2.2(a)] shows that $f(x) \rightarrow 2$ as $x \rightarrow 1$. To evaluate the limit of $f(x)$ as $x \rightarrow 1$, we write ${ }^{3}$

This approaches 2 as $x$ approaches 1 , hence

So, while $f(1)$ is not defined ( $x=1$ is not in the domain and $y=2$ is not in the range of $f$ ) the $\operatorname{limit}_{x \rightarrow 1} f(x)$ exists. This is not always the case of course. As another example consider the function

and suppose we want to evaluate $\lim _{x \rightarrow 1} f(x)$ as before. We can see from the graph of the function in Fig. 2.2(b) that the limit does not exist as the function has a vertical asymptote at $x=1$ (shown by a vertical, dotted line). In order for a limit to exist, the function must be settling down to a single finite value as we approach the point.

Limits can be taken as $x \rightarrow \infty$ or $x \rightarrow-\infty$. Some examples are given next.

${ }^{3}$ L'Hôpital's rule (see Section 2.3) can also be used here.

(a)

(b)

Figure 2.2: (a) The graph of a function $y=f(x)$ where $f(x)=\left(x^{2}-1\right) /(x-1)$. The circle marker at $x=1, y=2$ represents the point in $x$ at which the function is not defined. The point $x=1$ is not in the domain of $f$; further, the point $y=2$ is not in the range of $f$. The marked point is the limit given by $\lim _{x \rightarrow 1} f(x)$. The limit as $x \rightarrow 1$ is finite even though $f(1)$ is undefined. (b) The graph of a function $y=f(x)$ where $f(x)=\left(x^{2}+1\right) /(x-1)$. The function is not defined at $x=1$ and the $\operatorname{limit}^{\lim _{x \rightarrow 1} f(x)}$ does not exist. The dotted line at $x=1$ is a vertical asymptote.

Note that for the last function above, the limit is obvious from Fig. 1.3(b) which shows that the range of $y=\tan ^{-1}(x)$ is $-\pi / 2<x<\pi / 2$. The limit $x \rightarrow \infty$ is the same as letting $1 / x \rightarrow 0^{+}$; therefore one can transform an infinite limit to a non-infinite limit, as follows:

where $s=1 / x$.

One-sided

Consider the function $y=H(x)$ given by

with its graph shown in Fig. 2.3 for $x \in[-4,4]$. This function is a piecewise-defined function called the Heaviside function or step function often denoted by $H_{0}(x)$. The function has a jump at $x=0$ and is said to be discontinuous at $x=0$ (for more details on continuity, see Section 2.4). Recall from Chapter 1, that a function must be single-valued, i.e. each point in $\mathcal{D}$ is sent to a uniquely defined point in $\mathcal{R}$. For this reason we have $H(x)=0$ for $x<0$ and $H(x)=1$ for $x \geq 0$.

Now suppose we want to evaluate the limit,

We can see from the graph that if $x \rightarrow 0^{-}$i.e. if 0 is approached from the left, $y \rightarrow 0$ while, if $x \rightarrow 0^{+}, y \rightarrow 1$ (note how in the limit given by (2.6), we used $x \rightarrow 0$ as opposed to $x \rightarrow 0^{ \pm}$). However, the function needs to approach a single value when the point (here, this is 0$)$ is approached from both sides. Therefore, the $\varlimsup$ limit (2.6) does not exist or, more particularly, the two-sided limit does not exist. It is possible to define one-sided limits; the right-handed limit

says the limit is $L$ for all $x$ sufficiently close to $a$ with $x>a$. The left-handed limit

says the limit is $L$ for all $x$ sufficiently close to $a$ with $x<a$. For the function in Eq. $(2.5)$,

Figure 2.3: A graph of the Heaviside function $y=H(x)$ given by Eq. (2.5). A solid marker indicates that the endpoint $x=0$ is included while a hollow marker indicates that it is not included.

Properties

Here we outline several rules that hold for limits based on the assumption that $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ exist.

I. $\lim _{x \rightarrow a}[c f(x)]=c \lim _{x \rightarrow a} f(x)$, where $c$ is a constant.

II. $\lim _{x \rightarrow a}[f(x) \pm g(x)]=\lim _{x \rightarrow a} f(x) \pm \lim _{x \rightarrow a} g(x)$.

III. $\lim _{x \rightarrow a}[f(x) g(x)]=\lim _{x \rightarrow a} f(x) \lim _{x \rightarrow a} g(x)$.

IV. $\lim _{x \rightarrow a}\left[\frac{f(x)}{g(x)}\right]=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}$, provided $\lim _{x \rightarrow a} g(x) \neq 0$. V. $\lim _{x \rightarrow a}[f(x)]^{n}=\left[\lim _{x \rightarrow a} f(x)\right]^{n}$, where $n$ is any real number. For integer $n$, this is a generalisation of the product rule given by Property III.

We also have that $\lim _{x \rightarrow a} c=c$, i.e. the limit of a constant is just the constant and $\lim _{x \rightarrow a} x=a$; these results are easy to see by plotting the graphs of the functions $f(x)=c$ and $f(x)=x$.