Limit of functions – "Math for Non-Geeks"

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A New Attempt with a Rough Plan

Intuition

We have an arbitrary function $f$. This article deals with the question "How does $f$ behave in the neighborhood of a point $x_0$, or near infinity?" And "Does $f$ tends to a particular value as we approach $x_0$ along the x-axis, or does it continue on to infinity?"

We will consider three example functions at the origin:

First Example

$f:ℝ\to ℝ,f(x)=\exp (x)$

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  Convergence of $f(x)=\exp (x)$ as $x$ approaches $0$ from the right
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  Convergence of $f(x)=\exp (x)$ as $x$ approaches $0$ from the left

Regardless of who we approach $x_0=0$ along the x-axis, $f(x)$ tends towards $1$.

Second Example

$f:ℝ\setminus \{0\}\to ℝ,f(x)=\{\begin{array}{cc}1& \text{if }x<0\\ x+1& \text{if }x>0\end{array}$

The function f

Even though $f$ is not defined at $0$, the function still tends towards the value $1$ at the point $0$.

Third Example

$f:ℝ\to ℝ,f(x)=\{\begin{array}{cc}1& \text{if }x>0\\ 0,& \text{if }x=0\\ -1& \text{if }x<0\end{array}$

The sign function \sgn

This case is not as easy as the previous two. From the left $f$ tends towards $-1$, from the right towards $1$. If we assign the functional value of $x_0=0$ to the value of $x_0$ itself, i.e set $f(x_0)=0$, then $f$ can jump back and forth between $-1$ and $0$ as well as between $1$ and $0$.

Application Examples

Limits at infinity:

In order to produce flash on a camera, capacitors are discharged within fractions of a second. Physically, the discharge of a capacitor can be written as $V(t)=V_0\cdot e^{-\frac{t}{R\cdot C}}$.

Discharging of a capacitor

For a positive initial voltage $V_0$ it doesn't matter how large or small we choose the time unit $t$. It holds $V(t)>0$ and in particular $V(t)\ne 0$. How can we mathematically express that the voltage and therefore also the charge of the capacitor approaches $0$? To this we have to investigate $\underset{t\to \mathrm{\infty }}{lim}V(t)$, i.e. the limit of $V$ at infinity.

$x_0$ as a real number:

In introductory calculus, the area under the graph of a function on the interval $[a,b]$ is often approximated by the area of rectangles of equal width.

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The thinner the rectangles, the more precisely they approximate the area under the curve. If we write $\mathrm{\Delta }x$ for the width of a rectangle, we know that the area of a rectangle is the width times the height. The height of the rectangle is precisely the functional value of $f$ at the edge of the rectangle. For rectangle width $\mathrm{\Delta }x$ we can calculate an approximation of the area under the curve of $f$ on the interval $[a,b]$ by the formula: $g(\mathrm{\Delta }x)=\sum _i^{}\mathrm{\Delta }x\ast f(a+i\ast \mathrm{\Delta }x)$, as long as we make sure the boundaries of summation fit the index $i$. We can also calculate and give an explicit formula for the functional values of the function $g$ at the arguments $\mathrm{\Delta }x>0$ since we don't allow rectangles to have a width $\le 0$. In essence, we have formulated the integral calculation as a problem wherein we want to find the limit of $g$ in $x_0=0$.

Transition to Mathematics

How can we as humans consider how a function "behaves" near a point? E.g. does the function increase? Decrease? Go to infinity? Have a hole or jump discontinuity? Is the point a minimum or maximum? In typical introductory math courses, a simple method is to choose a few points near a given point $x_0$ and compute their functional values to obtain some relative idea or model of how the function looks. Now we want to formalize this procedure:

Let's consider a sequence $(x_n{)}_{n\in \mathbb{N}}$ and substitute the sequence elements $x_n$ into $f$. Since these points should approach $x_0$, let's consider functions for which $\underset{n\to \mathrm{\infty }}{lim}{x}_n=x_0$ holds. For example, it would be unwise to look at $f$ at the point $x_0=1$ but to use test values like $10,100,1000,\mathrm{...}$.

By using the $x_n$'s as our arguments that we will set into the function, this yields yet another sequence $(f(x_n){)}_{n\in \mathbb{N}}$. We ask whether $f$ converges to a functional value in $x_0$ (remember if $f$ goes to infinity at the point $x_0$ then this sequence does not converge to a functional value). I.e. this is the same as asking whether $\underset{n\to \mathrm{\infty }}{lim}f(x_n)$ exists.

We haven't yet discussed how many sequences we have to set into $f$. Do we get enough information if we only choose one sequence and observe how it behaves as it tends towards $x_0$?. Let's consider the following example:

The sign function is given by $\mathrm{sgn}:ℝ\to ℝ,\mathrm{sgn}(x)=\{\begin{array}{cc}1& x>0\\ 0& x=0\\ -1& x<0\end{array}$

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  Approximation with the sequence $\frac{1}{n}$ (please click)
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  Approximation with the sequence $-\frac{1}{n}$ (please click)

If we choose the sequence $(x_n{)}_{n\in \mathbb{N}}$ with $x_n=\frac{1}{n}$ and substitute this into $\mathrm{sgn}$, then we always get the value $\mathrm{sgn}(x_n)=1$. If we only looked at this one sequence, then we would assume that $\mathrm{sgn}$ converges to the value of 1 at the point $x_0=0$. If we choose the sequence $(x_n{)}_{n\in \mathbb{N}}$ with $x_n=-\frac{1}{n}$, then we always get the value $\mathrm{sgn}(x_n)=-1$ and now we see that the function $\mathrm{sgn}$ doesn't have a unique limit at the point $x_0=0$. So it is not sufficient to consider just one sequence. Thus, the function has to be the same for all possible sequences, so that we can determine the existence of the limit value.

Definition by Sequences

Definition (Limit of a Sequence)

Let $f:D\to ℝ$ be a function with $D\subseteq ℝ$, $a\in ℝ$ and $x_0\in \bar{D}$. $f$ has a limit $a$ at the point $x_0$ , if for every sequence $(x_n{)}_{n\in \mathbb{N}}$ with $\mathrm{\forall }n\in \mathbb{N}:x_n\in D$ and $\underset{n\to \mathrm{\infty }}{lim}{x}_n=x_0$ it holds: $\underset{n\to \mathrm{\infty }}{lim}f(x_n)=a$. If this is the case, then we write $\underset{x\to x_0}{lim}f(x)=a$

Let $f:D\to ℝ$ be a function with $D\subseteq ℝ$ and $a\in ℝ$. $f$ has the limit value $a$ at infinity , if for every sequence $(x_n{)}_{n\in \mathbb{N}}$ with $\mathrm{\forall }n\in \mathbb{N}:x_n\in D$ and $\underset{n\to \mathrm{\infty }}{lim}{x}_n=\mathrm{\infty }$ it holds: $\underset{n\to \mathrm{\infty }}{lim}f(x_n)=a$. If this is the case, we write $\underset{x\to \mathrm{\infty }}{lim}f(x)=a$

Brief Addendum on the Definition of Sequences

As a final touch, we had to note that the expression $f(x_n)$ only makes sense if $x_n$ lies within the domain of $f$. Therefore, we require $\mathrm{\forall }n\in \mathbb{N}:x_n\in D$. We should also only examine $f$ at points $x_0$ that we can actually approach. If, for example, $f$ is only defined on $D=[0,1)$, then we cannot determine what $f$ does in the vicinity of $x_0=5$. What is possible, however, is to ask for an approximation of $f$ at the point $x_0=1$. Our $x_0$ must therefore lie in $\bar{D}$, that is, the closure of $D$.

Variations of the Definition

This section exists because different authors and different textbooks use different definitions. It may happen that you did not learn the definition from this article in your lecture, but one of the following variations.

Instead of $x_0\in \bar{D}$, it is possible to allow only $x_0\in D$.

Regardless of the first decision, it is possible to further restrict the sequences $(x_n{)}_{n\in \mathbb{N}}$ under consideration. If we want to consider the behaviour of $f$ in the ‘vicinity’ of $x_0\in D$, we can discuss whether it is permissible to use $x_0$ itself as a value ‘close to $x_0$’ in $f$. Some authors therefore require:

For the sequences $(x_n{)}_{n\in \mathbb{N}}$ under consideration, $x_n\in D\setminus \{x_0\}$ $\mathrm{\forall }n\in \mathbb{N}$ (instead of $x_n\in D$)

Note: Continuity and limits of functions

If we compare our intuition with the sequence criterion of continuity, we see that continuity also involves the convergence of $f$ at a point $x_0$. If we also compare both definitions via sequences, we find very similar results here too. For the continuity of $f$ at the point $x_0$, the only additional requirement is that $x_0\in D$ must hold, because the expression $f(x_0)$ must exist. (Since $D\subset \bar{D}$, in particular $x_0\in \bar{D}$ holds.) In fact, there are authors who define continuity using limits of functions. Continuity of $f$ at $x_0\in D$ means that $\underset{x\to x_0}{lim}f(x)=f(x_0)$. This is independent of which version we choose in the definition via sequences. In the first version, this is because we have to impose the restriction $x_0\in D$ for the question of continuity anyway. It also does not matter whether we require $x_n\in D\setminus \{x_0\}$ or $x_n\in D$: If, for all permitted sequences in $D\setminus \{x_0\}$, $f(x_n)$ tends towards $f(x_0)$, then this also applies to all permitted sequences in $D$. Conversely, if for all permitted sequences in $D$, $f(x_n)$ approaches $f(x_0)$, then this also applies to all sequences in $D\setminus \{x_0\}$.

It should also be noted that the definition of continuity via limits of functions also works with the epsilon-delta definition in this chapter.

Use of one-sided limits

"Later for integrals: space of regular functions. Regular function is relatively abstract. Classifiable as: f regular function if for all $x\in D$ right-sided and left-sided limits exist.