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Question Number 197832 by mathlove last updated on 30/Sep/23

lim_(x→∞)  ((2+(√(cosx)))/(−1+(√(cosx))))=?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{2}+\sqrt{{cosx}}}{−\mathrm{1}+\sqrt{{cosx}}}=? \\ $$

Answered by MM42 last updated on 30/Sep/23

let  f(x)=((2+(√(cosx)))/(−1+(√(cosx))))  for  a_n =2nπ⇒lim_(n→∞)  f(a_n )=∞  for   b_n =2nπ+(π/2)⇒lim_(n→∞) f(b_n )=−2  ⇒lim : not exist

$${let}\:\:{f}\left({x}\right)=\frac{\mathrm{2}+\sqrt{{cosx}}}{−\mathrm{1}+\sqrt{{cosx}}} \\ $$$${for}\:\:{a}_{{n}} =\mathrm{2}{n}\pi\Rightarrow{lim}_{{n}\rightarrow\infty} \:{f}\left({a}_{{n}} \right)=\infty \\ $$$${for}\:\:\:{b}_{{n}} =\mathrm{2}{n}\pi+\frac{\pi}{\mathrm{2}}\Rightarrow{lim}_{{n}\rightarrow\infty} {f}\left({b}_{{n}} \right)=−\mathrm{2} \\ $$$$\Rightarrow{lim}\::\:{not}\:{exist} \\ $$

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