Skip to main content
Mathematics LibreTexts

4.2.E: More Problems on Limits and Continuity

  • Page ID
    22631
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Exercise \(\PageIndex{1}\)

    Complete the proof of Corollary 1.
    \(\left[\text { Hint: Consider }\left\{f\left(x_{m}\right)\right\} \text { and }\left\{f\left(x_{m}^{\prime}\right)\right\}, \text { with }\right.\)
    \[
    x_{m} \rightarrow p \text { and } x_{m}^{\prime} \rightarrow p .
    \]
    By Chapter 3, §14, Corollary \(4, p\) is also the limit of
    \[
    x_{1}, x_{1}^{\prime}, x_{2}, x_{2}^{\prime}, \ldots ,
    \]
    so, by assumption,
    \[
    f\left(x_{1}\right), f\left(x_{1}^{\prime}\right), \ldots \text { converges (to } q, \text { say } ) .
    \]
    Hence \(\left\{f\left(x_{m}\right)\right\}\) and \(\left\{f\left(x_{m}^{\prime}\right)\right\}\) must have the same limit \(q .(\mathrm{Why} ?) ]\)

    Exercise \(\PageIndex{*2}\)

    Complete the converse proof of Theorem 2 (cf. proof of Theorem 1 in
    Chapter 3, §17).

    Exercise \(\PageIndex{3}\)

    Define \(f, g : E^{1} \rightarrow E^{1}\) by setting
    (i) \(f(x)=2 ; g(y)=3\) if \(y \neq 2,\) and \(g(2)=0 ;\) or
    (ii) \(f(x)=2\) if \(x\) is rational and \(f(x)=2 x\) otherwise; \(g\) as in \((\mathrm{i})\).
    In both cases, show that
    \[
    \lim _{x \rightarrow 1} f(x)=2 \text { and } \lim _{y \rightarrow 2} g(y)=3 \text { but not } \lim _{x \rightarrow 1} g(f(x))=3 .
    \]

    Exercise \(\PageIndex{4}\)

    Prove Theorem 3 from "\(\varepsilon, \delta\) " definitions. Also prove (both ways) that if \(f\) is relatively continuous on \(B,\) and \(g\) on \(f[B],\) then \(g \circ f\) is relatively continuous on \(B\).

    Exercise \(\PageIndex{5}\)

    Complete the missing details in Examples (A) and (B).
    [Hint for (B): Verify that
    \[
    \left(1-\frac{1}{n+1}\right)^{-n-1}=\left(\frac{n}{n+1}\right)^{-n-1}=\left(\frac{n+1}{n}\right)^{n+1}=\left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)^{n} \rightarrow e . ]
    \]

    Exercise \(\PageIndex{6}\)

    \(\Rightarrow 6 .\) Given \(f, g, h : A \rightarrow E^{*}, A \subseteq(S, \rho),\) with
    \[
    f(x) \leq h(x) \leq g(x)
    \]
    for \(x \in G_{\neg p}(\delta) \cap A\) for some \(\delta>0 .\) Prove that if
    \[
    \lim _{x \rightarrow p} f(x)=\lim _{x \rightarrow p} g(x)=q ,
    \]
    also then
    \[
    \lim _{x \rightarrow p} h(x)=q.
    \]
    Use Theorem 1.
    [Hint: Take any
    \[
    \left\{x_{m}\right\} \subseteq A-\{p\} \text { with } x_{m} \rightarrow p .
    \]
    Then \(f\left(x_{m}\right) \rightarrow q, g\left(x_{m}\right) \rightarrow q,\) and
    \[
    \left(\forall x_{m} \in A \cap G_{\neg p}(\delta)\right) \quad f\left(x_{m}\right) \leq h\left(x_{m}\right) \leq g\left(x_{m}\right) .
    \]
    Now apply Corollary 3 of Chapter \(3, §15 . ]\)

    Exercise \(\PageIndex{7}\)

    \(\Rightarrow 7 .\) Given \(f, g : A \rightarrow E^{*}, A \subseteq(S, \rho),\) with \(f(x) \rightarrow q\) and \(g(x) \rightarrow r\) as \(x \rightarrow p\) \((p \in S),\) prove the following:
    (i) If \(q>r,\) then
    \[
    (\exists \delta>0)\left(\forall x \in A \cap G_{\neg p}(\delta)\right) \quad f(x)>g(x) .
    \]
    (ii) (Passage to the limit in inequalities.) If
    \[
    (\forall \delta>0)\left(\exists x \in A \cap G_{\neg p}(\delta)\right) \quad f(x) \leq g(x) ,
    \]
    then \(q \leq r .\) (Observe that here \(A\) clusters at \(p\) necessarily, so the limits are unique.)
    [Hint: Proceed as in Problem \(6 ;\) use Corollary 1 of Chapter \(3, §15 .]\)

    Exercise \(\PageIndex{8}\)

    Do Problems 6 and 7 using only Definition 2 of §1.
    \([\text { Hint: Here prove } 7(\text { ii }) \text { first. }]\)

    Exercise \(\PageIndex{9}\)

    Do Examples \((a)-(d)\) of §1 using Theorem 1.
    [Hint: For \((\mathrm{c}),\) use also Example (a) in Chapter \(3, §16 . ]\)

    Exercise \(\PageIndex{10}\)

    Addition and multiplication in \(E^{1}\) may be treated as functions
    \[
    f, g : E^{2} \rightarrow E^{1}
    \]
    with
    \[
    f(x, y)=x+y \text { and } g(x, y)=x y .
    \]
    Show that \(f\) and \(g\) are continuous on \(E^{2}\) (see footnote 2 in Chapter 3 §15). Similarly, show that the standard metric
    \[
    \rho(x, y)=|x-y|
    \]
    is a continuous mapping from \(E^{2}\) to \(E^{1}\).
    \([\text { Hint: Use Theorems } 1,2, \text { and, } 4 \text { of Chapter } 3, §15 \text { and the sequential criterion. }]\)

    Exercise \(\PageIndex{11}\)

    Using Corollary 2 and formula \((9),\) find \(\lim _{x \rightarrow 0}(1 \pm m x)^{1 / x}\) for a fixed \(m \in N .\)

    Exercise \(\PageIndex{12}\)

    \(\Rightarrow 12 .\) Let \(a>0\) in \(E^{1} .\) Prove that \(\lim _{x \rightarrow 0} a^{x}=1\).
    \(\left[\text { Hint: Let } n=f(x) \text { be the integral part of } \frac{1}{x}(x \neq 0) . \text { Verify that }\right.\)
    \[
    a^{-1 /(n+1)} \leq a^{x} \leq a^{1 / n} \text { if } a \geq 1 ,
    \]
    with inequalities reversed if \(0<a<1 .\) Then proceed as in Example \((\mathrm{A}),\) noting that
    \[
    \lim _{n \rightarrow \infty} a^{1 / n}=1=\lim _{n \rightarrow \infty} a^{-1 /(n+1)}
    \]
    by Problem 20 of Chapter \(3, §15 .(\text { Explain! })]\)

    Exercise \(\PageIndex{13}\)

    \(\Rightarrow 13 .\) Given \(f, g : A \rightarrow E^{*}, A \subseteq(S, \rho),\) with
    \[
    f \leq g \quad \text { for } x \text { in } G_{\neg p}(\delta) \cap A .
    \]
    Prove that
    (a) if \(\lim _{x \rightarrow p} f(x)=+\infty,\) then also \(\lim _{x \rightarrow p} g(x)=+\infty\);
    (b) if \(\lim _{x \rightarrow p} g(x)=-\infty,\) then also \(\lim _{x \rightarrow p} f(x)=-\infty\).
    Do it it two ways:
    (i) Use definitions only, such as \(\left(2^{\prime}\right)\) in \(§1\).
    (ii) Use Problem 10 of Chapter 2, §13 and the sequential criterion.

    Exercise \(\PageIndex{14}\)

    \(\Rightarrow 14 .\) Prove that
    (i) if \(a>1\) in \(E^{1},\) then
    \[
    \lim _{x \rightarrow+\infty} \frac{a^{x}}{x}=+\infty \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x}=0 ;
    \]
    (ii) if \(0<a<1,\) then
    \[
    \lim _{x \rightarrow+\infty} \frac{a^{x}}{x}=0 \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x}=+\infty ;
    \]
    (iii) if \(a>1\) and \(0 \leq q \in E^{1},\) then
    \[
    \lim _{x \rightarrow+\infty} \frac{a^{x}}{x^{q}}=+\infty \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x^{q}}=0 ;
    \]
    \((\text { iv })\) if \(0<a<1\) and \(0 \leq q \in E^{1},\) then
    \[
    \lim _{x \rightarrow+\infty} \frac{a^{x}}{x^{q}}=0 \text { and } \lim _{x \rightarrow+\infty} \frac{a^{-x}}{x^{q}}=+\infty .
    \]
    [Hint: (i) From Problems 17 and 10 of Chapter \(3, §15,\) obtain
    \[
    \lim \frac{a^{n}}{n}=+\infty .
    \]
    Then proceed as in Examples \((\mathrm{A})-(\mathrm{C}) ;\) (iii) reduces to (i) by the method used in
    Problem 18 of Chapter \(3, §15 . ]\)

    Exercise \(\PageIndex{15}\)

    \(\Rightarrow * 15 .\) For a map \(f :(S, \rho) \rightarrow\left(T, \rho^{\prime}\right),\) show that the following statements are equivalent:
    (i) \(f\) is continuous on \(S\).
    (ii) \((\forall A \subseteq S) f[\overline{A}] \subseteq \overline{f[A]}\).
    (iii) \((\forall B \subseteq T) f^{-1}[\overline{B}] \supseteq \overline{f^{-1}[B]}\).
    (iv) \(f^{-1}[B]\) is closed in \((S, \rho)\) whenever \(B\) is closed in \(\left(T, \rho^{\prime}\right)\).
    (v) \(f^{-1}[B]\) is open in \((S, \rho)\) whenever \(B\) is open in \(\left(T, \rho^{\prime}\right)\).
    [Hints: \( (i) \Longrightarrow(\mathrm{ii})\): Use Theorem 3 of Chapter } 3, §16 and the sequential criterion to show that
    \[
    p \in \overline{A} \Longrightarrow f(p) \in \overline{f[A]} .
    \]
    (ii) \(\Longrightarrow(\text { iii }) :\) Let \(A=f^{-1}[B] .\) Then \(f[A] \subseteq B,\) so by \((\text { ii })\),
    \[
    f[\overline{A}] \subseteq \overline{f[A]} \subseteq \overline{B} .
    \]
    Hence
    \[
    \overline{f^{-1}[B]}=\overline{A} \subseteq f^{-1}[f[\overline{A}]] \subseteq f^{-1}[\overline{B}] . \quad \text { (Why?) }
    \]
    (iii) \(\Longrightarrow(\mathrm{iv}) :\) If \(B\) is closed, \(B=\overline{B}\) (Chapter 3, §16, Theorem 4(ii)), so by (iii),
    \[
    f^{-1}[B]=f^{-1}[\overline{B}] \supseteq \overline{f^{-1}[B]} ; \text { deduce (iv) } .
    \]
    \((\mathrm{iv}) \Longrightarrow(\mathrm{v}) :\) Pass to complements in \((\mathrm{iv})\).
    \((\mathrm{v}) \Longrightarrow(\mathrm{i}) :\) Assume \((\mathrm{v}) .\) Take any \(p \in S\) and use Definition 1 in \(§1 . ]\)

    Exercise \(\PageIndex{16}\)

    Let \(f : E^{1} \rightarrow E^{1}\) be continuous. Define \(g : E^{1} \rightarrow E^{2}\) by
    \[
    g(x)=(x, f(x)) .
    \]
    Prove that
    (a) \(g\) and \(g^{-1}\) are one to one and continuous;
    (b) the range of \(g,\) i.e., the set
    \[
    D_{g}^{\prime}=\left\{(x, f(x)) | x \in E^{1}\right\} ,
    \]
    is closed in \(E^{2}\).
    [Hint: Use Theorem 2 of Chapter 3, §15, Theorem 4 of Chapter 3, §16, and the sequential criterion.]


    4.2.E: More Problems on Limits and Continuity is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?