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Question Number 115384 by mathdave last updated on 25/Sep/20
solve the limit problem  lim_(x→∞) (((cosx)/(1−e^(−x) )))
$${solve}\:{the}\:{limit}\:{problem} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{cos}{x}}{\mathrm{1}−{e}^{−{x}} }\right) \\ $$
Answered by MJS_new last updated on 25/Sep/20
lim_(x→∞)  ((cos x)/(1−e^(−x) )) =lim_(x→∞)  (e^x /(e^x −1))cos x =lim_(x→∞)  cos x  ⇒ limit doesn′t exist
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{cos}\:{x}}{\mathrm{1}−\mathrm{e}^{−{x}} }\:=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{e}^{{x}} }{\mathrm{e}^{{x}} −\mathrm{1}}\mathrm{cos}\:{x}\:=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{cos}\:{x} \\ $$$$\Rightarrow\:\mathrm{limit}\:\mathrm{doesn}'\mathrm{t}\:\mathrm{exist} \\ $$

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