If $lim f(x) = 0,$ then $lim 1/f(x) = infty.$












0












$begingroup$



Suppose that $f:DtoBbb R$, where $D$ is a subset of $Bbb R$ and $a$ is an accumulation point of $D$, $lim_{xto a+}f(x)=0$, and $f(x)ne0$ for any $x$ in $D$ in some neighborhood of $a$.
I understand from the following proof that if $lim f(x) = 0,$ then $lim 1/|f(x)| = infty.$ : proof for |f(x)|,
however, does it also apply for $lim 1/f(x)= infty$ ? or $lim 1/f(x)= -infty$ If not then why?











share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    What if $f$ only takes negative values?
    $endgroup$
    – José Carlos Santos
    Dec 5 '18 at 11:21






  • 2




    $begingroup$
    the problem is whether you consider $infty$ to be either $pm infty$ or only $+infty$
    $endgroup$
    – TheD0ubleT
    Dec 5 '18 at 11:22












  • $begingroup$
    What if $f$ takes both positive and negative values ($f$ would not be continuous)?
    $endgroup$
    – Michael Burr
    Dec 5 '18 at 11:23










  • $begingroup$
    @MichaelBurr how can f take both of those, conjure to 0 and then not be continuous? when it's $lim_{xto a+}$?
    $endgroup$
    – Yuki1112
    Dec 5 '18 at 11:25












  • $begingroup$
    @TheD0ubleT either, sorry, I edited it to be more clear on the matter
    $endgroup$
    – Yuki1112
    Dec 5 '18 at 11:31
















0












$begingroup$



Suppose that $f:DtoBbb R$, where $D$ is a subset of $Bbb R$ and $a$ is an accumulation point of $D$, $lim_{xto a+}f(x)=0$, and $f(x)ne0$ for any $x$ in $D$ in some neighborhood of $a$.
I understand from the following proof that if $lim f(x) = 0,$ then $lim 1/|f(x)| = infty.$ : proof for |f(x)|,
however, does it also apply for $lim 1/f(x)= infty$ ? or $lim 1/f(x)= -infty$ If not then why?











share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    What if $f$ only takes negative values?
    $endgroup$
    – José Carlos Santos
    Dec 5 '18 at 11:21






  • 2




    $begingroup$
    the problem is whether you consider $infty$ to be either $pm infty$ or only $+infty$
    $endgroup$
    – TheD0ubleT
    Dec 5 '18 at 11:22












  • $begingroup$
    What if $f$ takes both positive and negative values ($f$ would not be continuous)?
    $endgroup$
    – Michael Burr
    Dec 5 '18 at 11:23










  • $begingroup$
    @MichaelBurr how can f take both of those, conjure to 0 and then not be continuous? when it's $lim_{xto a+}$?
    $endgroup$
    – Yuki1112
    Dec 5 '18 at 11:25












  • $begingroup$
    @TheD0ubleT either, sorry, I edited it to be more clear on the matter
    $endgroup$
    – Yuki1112
    Dec 5 '18 at 11:31














0












0








0





$begingroup$



Suppose that $f:DtoBbb R$, where $D$ is a subset of $Bbb R$ and $a$ is an accumulation point of $D$, $lim_{xto a+}f(x)=0$, and $f(x)ne0$ for any $x$ in $D$ in some neighborhood of $a$.
I understand from the following proof that if $lim f(x) = 0,$ then $lim 1/|f(x)| = infty.$ : proof for |f(x)|,
however, does it also apply for $lim 1/f(x)= infty$ ? or $lim 1/f(x)= -infty$ If not then why?











share|cite|improve this question











$endgroup$





Suppose that $f:DtoBbb R$, where $D$ is a subset of $Bbb R$ and $a$ is an accumulation point of $D$, $lim_{xto a+}f(x)=0$, and $f(x)ne0$ for any $x$ in $D$ in some neighborhood of $a$.
I understand from the following proof that if $lim f(x) = 0,$ then $lim 1/|f(x)| = infty.$ : proof for |f(x)|,
however, does it also apply for $lim 1/f(x)= infty$ ? or $lim 1/f(x)= -infty$ If not then why?








limits






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 5 '18 at 11:23







Yuki1112

















asked Dec 5 '18 at 11:18









Yuki1112Yuki1112

174




174








  • 3




    $begingroup$
    What if $f$ only takes negative values?
    $endgroup$
    – José Carlos Santos
    Dec 5 '18 at 11:21






  • 2




    $begingroup$
    the problem is whether you consider $infty$ to be either $pm infty$ or only $+infty$
    $endgroup$
    – TheD0ubleT
    Dec 5 '18 at 11:22












  • $begingroup$
    What if $f$ takes both positive and negative values ($f$ would not be continuous)?
    $endgroup$
    – Michael Burr
    Dec 5 '18 at 11:23










  • $begingroup$
    @MichaelBurr how can f take both of those, conjure to 0 and then not be continuous? when it's $lim_{xto a+}$?
    $endgroup$
    – Yuki1112
    Dec 5 '18 at 11:25












  • $begingroup$
    @TheD0ubleT either, sorry, I edited it to be more clear on the matter
    $endgroup$
    – Yuki1112
    Dec 5 '18 at 11:31














  • 3




    $begingroup$
    What if $f$ only takes negative values?
    $endgroup$
    – José Carlos Santos
    Dec 5 '18 at 11:21






  • 2




    $begingroup$
    the problem is whether you consider $infty$ to be either $pm infty$ or only $+infty$
    $endgroup$
    – TheD0ubleT
    Dec 5 '18 at 11:22












  • $begingroup$
    What if $f$ takes both positive and negative values ($f$ would not be continuous)?
    $endgroup$
    – Michael Burr
    Dec 5 '18 at 11:23










  • $begingroup$
    @MichaelBurr how can f take both of those, conjure to 0 and then not be continuous? when it's $lim_{xto a+}$?
    $endgroup$
    – Yuki1112
    Dec 5 '18 at 11:25












  • $begingroup$
    @TheD0ubleT either, sorry, I edited it to be more clear on the matter
    $endgroup$
    – Yuki1112
    Dec 5 '18 at 11:31








3




3




$begingroup$
What if $f$ only takes negative values?
$endgroup$
– José Carlos Santos
Dec 5 '18 at 11:21




$begingroup$
What if $f$ only takes negative values?
$endgroup$
– José Carlos Santos
Dec 5 '18 at 11:21




2




2




$begingroup$
the problem is whether you consider $infty$ to be either $pm infty$ or only $+infty$
$endgroup$
– TheD0ubleT
Dec 5 '18 at 11:22






$begingroup$
the problem is whether you consider $infty$ to be either $pm infty$ or only $+infty$
$endgroup$
– TheD0ubleT
Dec 5 '18 at 11:22














$begingroup$
What if $f$ takes both positive and negative values ($f$ would not be continuous)?
$endgroup$
– Michael Burr
Dec 5 '18 at 11:23




$begingroup$
What if $f$ takes both positive and negative values ($f$ would not be continuous)?
$endgroup$
– Michael Burr
Dec 5 '18 at 11:23












$begingroup$
@MichaelBurr how can f take both of those, conjure to 0 and then not be continuous? when it's $lim_{xto a+}$?
$endgroup$
– Yuki1112
Dec 5 '18 at 11:25






$begingroup$
@MichaelBurr how can f take both of those, conjure to 0 and then not be continuous? when it's $lim_{xto a+}$?
$endgroup$
– Yuki1112
Dec 5 '18 at 11:25














$begingroup$
@TheD0ubleT either, sorry, I edited it to be more clear on the matter
$endgroup$
– Yuki1112
Dec 5 '18 at 11:31




$begingroup$
@TheD0ubleT either, sorry, I edited it to be more clear on the matter
$endgroup$
– Yuki1112
Dec 5 '18 at 11:31










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