Question Number 211944 by liuxinnan last updated on 25/Sep/24 $$ \\ $$$${f}\left({x}\right)=\frac{\sqrt{\mathrm{1}+{x}}−\sqrt{\mathrm{1}−{x}}}{\:\sqrt{\mathrm{1}+{x}}+\sqrt{\mathrm{1}−{x}}}\:\:\:\:{f}^{'} \left({x}\right)=? \\ $$$$ \\ $$$$ \\ $$ Answered by efronzo1 last updated on…
Question Number 211812 by mokys last updated on 21/Sep/24 $${prove}\:\underset{{x}\rightarrow\infty} {{lim}}\:\left(\:\mathrm{1}\:+\:\frac{\mathrm{5}}{{x}}\:\right)^{\frac{\mathrm{1}}{{x}}} −\:\mathrm{1}\:=\:\mathrm{5}\: \\ $$ Commented by mr W last updated on 22/Sep/24 $${wrong}! \\ $$$${the}\:{result}\:{should}\:{be}\:\mathrm{0}.…
Question Number 211753 by alcohol last updated on 19/Sep/24 $${find}\:{all}\:\left({n},{m}\right)\:{such}\:{that}\:\frac{{n}^{\mathrm{2}} −{m}}{{m}^{\mathrm{2}} −{n}}\:\in\:\mathbb{Z} \\ $$ Answered by Frix last updated on 20/Sep/24 $$\mathrm{There}\:\mathrm{are}\:\mathrm{infinitely}\:\mathrm{many}\:\mathrm{solutions}. \\ $$$$\mathrm{3}\:\mathrm{ways}\:\mathrm{to}\:\mathrm{find}\:\mathrm{some}: \\…
Question Number 211750 by Alijumaaxyz last updated on 19/Sep/24 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{{a}^{\mathrm{1}/{x}} +{b}^{\mathrm{1}/{x}} }{\mathrm{2}}\right)^{{x}} ;\left({a},{b}\right)\in\mathbb{R}_{+} ^{\ast} \\ $$ Answered by mehdee7396 last updated on 19/Sep/24 $${li}\underset{{x}\rightarrow\infty}…
Question Number 211632 by sonukgindia last updated on 15/Sep/24 Answered by A5T last updated on 15/Sep/24 $$\phi\left(\mathrm{1000}\right)=\mathrm{400} \\ $$$$\mathrm{2024}^{\mathrm{2024}} \equiv\mathrm{0}\left({mod}\:\mathrm{16}\right);\mathrm{2024}^{\mathrm{2024}} \equiv\mathrm{1}\left({mod}\:\mathrm{25}\right) \\ $$$$\Rightarrow\mathrm{2024}^{\mathrm{2024}} =\mathrm{25}{q}+\mathrm{1}\equiv\left(\mathrm{0}\:{mod}\:\mathrm{16}\right)\Rightarrow{q}\equiv\mathrm{7}\left({mod}\mathrm{16}\right) \\…
Question Number 211637 by CrispyXYZ last updated on 15/Sep/24 $${f}\left({x}\right)\:=\:\mathrm{sin}\:{x}\:−\:\mathrm{e}^{{x}} \:+\:\mathrm{1}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:{f}\left({x}\right)\:\mathrm{has}\:\mathrm{only}\:\mathrm{2}\:\mathrm{zeros}\:\mathrm{in}\:−\pi\leqslant{x}\leqslant\mathrm{0}. \\ $$ Answered by mehdee1342 last updated on 15/Sep/24 $${f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${f}\left(−\pi\right)=\mathrm{1}−{e}^{−\pi}…
Question Number 211633 by sonukgindia last updated on 15/Sep/24 Commented by Ghisom last updated on 15/Sep/24 $$\mathrm{I}\:\mathrm{think}\:\mathrm{it}'\mathrm{s}\:\frac{\mathrm{7}}{\mathrm{30}} \\ $$ Answered by MathematicalUser2357 last updated on…
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Question Number 211627 by MrGaster last updated on 15/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\mid\boldsymbol{{h}}\mid\mid_{\:^{\boldsymbol{{L}}^{\boldsymbol{{p}}} } } ^{\boldsymbol{{p}}} =\int_{\mathrm{0}} ^{\infty} \mid\boldsymbol{{h}}\left(\boldsymbol{{x}}\right)\mid^{\boldsymbol{{p}}} \boldsymbol{{dx}}. \\ $$…