the definition of limit

Section 4.1 the definition of limit

Let's think about a function \(f:A\to \mathbb{R}\text{,}\) for some subset \(A\subseteq \mathbb{R}\text{.}\) We'd like to approach some point \(c\in\mathbb{R}\text{,}\) and see what's happening to \(f(x)\) as we do so. The first issue is: we have to actually be able to get close to \(c\) while staying in the domain of \(f\text{.}\) So it makes sense to demand that \(c\in A'\text{,}\) i.e., \(c\) is a limit point for \(A\text{.}\)

Then it's just a matter of turning the words "force", "arbitrarily", "guarantee", "sufficiently", and "close" into symbols.

Definition 4.1.1.

Let \(f:A\to \mathbb{R}\) be a function defined on a subset of \(\mathbb{R}\) and \(c\) a limit point for \(A\text{.}\) We write

\begin{equation*} \displaystyle\lim_{x\to c}f(x)=L \end{equation*}

\begin{equation*} \forall \epsilon\gt 0,\ \exists \delta\gt 0:\ 0\lt\lvert x-c\rvert\lt \delta\Rightarrow \lvert f(x)-L\rvert\lt \epsilon\ \ \ . \end{equation*}

Checkpoint 4.1.2.

Identify which keywords among the words listed above correspond to the symbolic parts of this definition.

Checkpoint 4.1.3.

What if we decided to drop the requirement that \(c\) be a limit point for \(A\text{?}\)

A point of \(A\) which isn't a limit point for \(A\) is called isolated. (You should satisfy yourself that this is a reasonable name.)

Prove that if \(c\) is an isolated point of \(A\text{,}\) then for any \(f\) and any \(L\text{,}\)

\begin{equation*} \forall \epsilon\gt 0,\ \exists \delta\gt 0:\ 0\lt\lvert x-c\rvert\lt \delta\Rightarrow \lvert f(x)-L\rvert\lt \epsilon\text{.} \end{equation*}

Example 4.1.4.

\(\displaystyle \lim_{x\to 3} x^2=9\text{.}\)

Proposition 4.1.5.

If \(\displaystyle \lim_{x\to c}f(x)=L\text{,}\) then \(f\) is bounded nearby to \(c\text{.}\)

Proof.

This is essentially the same proof as Theorem 2.1.14.

Pick \(\epsilon=1\text{.}\) Then there is \(\delta\gt 0\) so that for all \(x\) with \(0\lt \lvert x-c\rvert\lt \delta\text{,}\) \(\lvert f(x)-L\rvert\lt 1\text{.}\) For such \(x\text{,}\) \(\lvert f(x)\rvert\leq 1+\lvert L\rvert\text{.}\) So for any \(x\in (c-\delta,c+\delta)\text{,}\) we have \(\lvert f(x)\rvert\leq \max\{1+\lvert L\rvert,\lvert f(c)\rvert\}\text{.}\)

Proposition 4.1.6. sequential characterization of function limits.

\(\displaystyle\lim_{x\to c}f(x)=L\) if and only if for any sequence \((x_n)_{n\in\mathbb{N}}\) with \(x_n\neq c\) and \(x_n\to c\text{,}\) \(f(x_n)\to L\text{.}\)

Proposition 4.1.7.

Suppose \(f,g:A\to \mathbb{R}\) and \(\lambda\in \mathbb{R}\text{.}\) If \(\displaystyle\lim_{x\to c}f(x)=L,\lim_{x\to c}g(x)=M\text{,}\) then

\(\displaystyle \displaystyle\lim_{x\to c}f(x)+g(x)=L+M\)
\(\displaystyle \displaystyle\lim_{x\to c}f(x)g(x)=LM\)
\(\displaystyle \displaystyle\lim_{x\to c}\lambda f(x)=\lambda L\)

Remark 4.1.8.

You can either use Proposition 4.1.6 to prove Proposition 4.1.7, using the fact that sequence limits play nice with algebra; or you can prove Proposition 4.1.7 directly using the ideas from the proofs of corresponding statements about sequence limits.

For more exercise, you should write both proofs up.

Corollary 4.1.9.

If \(p(x)=a_0 +a_1x+\cdots +a_n x^n\) is a polynomial function, then any limit can be computed by evaluation; that is,

\begin{equation*} \lim_{x\to c}p(x)=p(c)\ \ . \end{equation*}

Remark 4.1.10.

The property of polynomials described in Corollary 4.1.9 is actually quite rare. Most limits cannot be computed simply by evaluating the limitand.

Example 4.1.11.

Define a function \(\mathbb{R}\to{0,1}\) by:

\begin{equation*} \chi_{\mathbb{Q}}(x)=\begin{cases}1&\text{ if }x\in\mathbb{Q}\\0&\text{ if }x\notin \mathbb{Q}\end{cases}\ \ . \end{equation*}

This function has no limits.

Example 4.1.12.

Define a function \(\Psi:\mathbb{R}\to [0,1]\) by:

\begin{equation*} \Psi(x)=\begin{cases}\frac{1}{n}& \text{ if }x=\frac{m}{n}\text{ in lowest terms}\\0&\text{ if }x\notin\mathbb{Q} \end{cases}\ \ . \end{equation*}

Then \(\displaystyle\lim_{x\to 0}\Psi(x)=0\text{.}\)