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lim-x-x-cos-1-x-




Question Number 110260 by bemath last updated on 28/Aug/20
lim_(x→∞)  x cos ((1/x)) ?
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}\:\mathrm{cos}\:\left(\frac{\mathrm{1}}{{x}}\right)\:? \\ $$
Answered by john santu last updated on 28/Aug/20
we know that −1≤cos (1/x)≤1   so lim_(x→∞) xcos ((1/x))= ∞
$${we}\:{know}\:{that}\:−\mathrm{1}\leqslant\mathrm{cos}\:\frac{\mathrm{1}}{{x}}\leqslant\mathrm{1}\: \\ $$$${so}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}{x}\mathrm{cos}\:\left(\frac{\mathrm{1}}{{x}}\right)=\:\infty \\ $$
Commented by Her_Majesty last updated on 28/Aug/20
I don′t think that′s true  let x=(1/t)⇒lim_(t→0)  ((cost)/t)  but (1/t)<0 for t<0 and (1/t)>0 for t>0  ⇒ limit does not exist
$${I}\:{don}'{t}\:{think}\:{that}'{s}\:{true} \\ $$$${let}\:{x}=\frac{\mathrm{1}}{{t}}\Rightarrow{lim}_{{t}\rightarrow\mathrm{0}} \:\frac{{cost}}{{t}} \\ $$$${but}\:\frac{\mathrm{1}}{{t}}<\mathrm{0}\:{for}\:{t}<\mathrm{0}\:{and}\:\frac{\mathrm{1}}{{t}}>\mathrm{0}\:{for}\:{t}>\mathrm{0} \\ $$$$\Rightarrow\:{limit}\:{does}\:{not}\:{exist} \\ $$
Commented by john santu last updated on 28/Aug/20
no
$${no} \\ $$
Commented by john santu last updated on 28/Aug/20
Commented by john santu last updated on 28/Aug/20
clear lim_(x→∞)  x.cos ((1/x))=∞
$${clear}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}.\mathrm{cos}\:\left(\frac{\mathrm{1}}{{x}}\right)=\infty \\ $$
Commented by Her_Majesty last updated on 28/Aug/20
you are right. my conclusion is wrong because  we′re only interested in lim_(t→0^+ )
$${you}\:{are}\:{right}.\:{my}\:{conclusion}\:{is}\:{wrong}\:{because} \\ $$$${we}'{re}\:{only}\:{interested}\:{in}\:{lim}_{{t}\rightarrow\mathrm{0}^{+} } \\ $$

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