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

🔗

Prev Top Next