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Question Number 105868 by pticantor last updated on 01/Aug/20

             lim_(x→0) cos((((√x)−1)/x))=?????  please i need your help

$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\boldsymbol{{li}}\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{m}cos}}\left(\frac{\sqrt{\boldsymbol{{x}}}−\mathrm{1}}{\boldsymbol{{x}}}\right)=????? \\ $$$${pl}\boldsymbol{{ease}}\:\boldsymbol{{i}}\:\boldsymbol{{need}}\:\boldsymbol{{your}}\:\boldsymbol{{help}} \\ $$

Commented by Dwaipayan Shikari last updated on 01/Aug/20

−1≤f(x)≤1

$$−\mathrm{1}\leqslant{f}\left({x}\right)\leqslant\mathrm{1} \\ $$

Answered by Her_Majesty last updated on 01/Aug/20

(((√x)−1)/x)=(1/(√x))−(1/x); x→0^+   let x=(1/t)  ⇒  (√t)−t; t→+∞  lim_(t→+∞) (√t)−t doesn′t exist ⇒ the limit you  asked for doesn′t exist too

$$\frac{\sqrt{{x}}−\mathrm{1}}{{x}}=\frac{\mathrm{1}}{\sqrt{{x}}}−\frac{\mathrm{1}}{{x}};\:{x}\rightarrow\mathrm{0}^{+} \\ $$$${let}\:{x}=\frac{\mathrm{1}}{{t}} \\ $$$$\Rightarrow \\ $$$$\sqrt{{t}}−{t};\:{t}\rightarrow+\infty \\ $$$$\underset{{t}\rightarrow+\infty} {{lim}}\sqrt{{t}}−{t}\:{doesn}'{t}\:{exist}\:\Rightarrow\:{the}\:{limit}\:{you} \\ $$$${asked}\:{for}\:{doesn}'{t}\:{exist}\:{too} \\ $$

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