Question Number 105868 by pticantor last updated on 01/Aug/20
$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\boldsymbol{{li}}\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{m}cos}}\left(\frac{\sqrt{\boldsymbol{{x}}}−\mathrm{1}}{\boldsymbol{{x}}}\right)=????? \\ $$$${pl}\boldsymbol{{ease}}\:\boldsymbol{{i}}\:\boldsymbol{{need}}\:\boldsymbol{{your}}\:\boldsymbol{{help}} \\ $$
Commented by Dwaipayan Shikari last updated on 01/Aug/20
$$−\mathrm{1}\leqslant{f}\left({x}\right)\leqslant\mathrm{1} \\ $$
Answered by Her_Majesty last updated on 01/Aug/20
$$\frac{\sqrt{{x}}−\mathrm{1}}{{x}}=\frac{\mathrm{1}}{\:\sqrt{{x}}}−\frac{\mathrm{1}}{{x}};\:{x}\rightarrow\mathrm{0}^{+} \\ $$$${let}\:{x}=\frac{\mathrm{1}}{{t}} \\ $$$$\Rightarrow \\ $$$$\sqrt{{t}}−{t};\:{t}\rightarrow+\infty \\ $$$$\underset{{t}\rightarrow+\infty} {{lim}}\sqrt{{t}}−{t}\:{doesn}'{t}\:{exist}\:\Rightarrow\:{the}\:{limit}\:{you} \\ $$$${asked}\:{for}\:{doesn}'{t}\:{exist}\:{too} \\ $$