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Question Number 211627 by MrGaster last updated on 15/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\mid\boldsymbol{{h}}\mid\mid_{\:^{\boldsymbol{{L}}^{\boldsymbol{{p}}} } } ^{\boldsymbol{{p}}} =\int_{\mathrm{0}} ^{\infty} \mid\boldsymbol{{h}}\left(\boldsymbol{{x}}\right)\mid^{\boldsymbol{{p}}} \boldsymbol{{dx}}. \\ $$…
Question Number 211578 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{+\infty} \frac{\boldsymbol{{x}}}{\:\sqrt{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{4}} }}\boldsymbol{{dx}}. \\ $$$$ \\ $$ Answered by lepuissantcedricjunior last updated on…
Question Number 211564 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}:\underset{\boldsymbol{{x}}=\mathrm{1}} {\overset{\mathrm{90}} {\sum}}\frac{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{{x}}^{\mathrm{2}} \right)}{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{{x}}\right)}=\mathrm{45} \\ $$$$ \\ $$$$ \\ $$ Commented by MrGaster last…
Question Number 211570 by liuxinnan last updated on 13/Sep/24 $$\left\{{x}_{{n}} \right\}>\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} ^{\frac{\mathrm{1}}{{n}}} ={a} \\ $$$${prove}\:{when}\:{a}<\mathrm{1}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}} =\mathrm{0} \\ $$$${when}\:{a}>\mathrm{1}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}_{{n}}…
Question Number 211567 by liuxinnan last updated on 13/Sep/24 $${if}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{g}\left({x}\right)=\mathrm{0} \\ $$$${when}\:{do}\:{not}\:{use}\:{f}\left({x}\right)\:{to}\:\:{replace}\:{g}\left({x}\right)\:\:\: \\ $$ Commented by MrGaster last updated on 13/Sep/24 $$\mathrm{1}.\mathrm{when}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({x}\right)=\underset{{x}\rightarrow\mathrm{0}}…
Question Number 211566 by MrGaster last updated on 13/Sep/24 $$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\underset{\boldsymbol{{n}}=−\infty} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{4}} +\boldsymbol{{a}}^{\mathrm{4}} }=\frac{\boldsymbol{\pi}}{\:\sqrt{\mathrm{2}}\boldsymbol{{a}}^{\mathrm{3}} }\:\frac{\boldsymbol{\mathrm{si}{n}\mathrm{h}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)+\boldsymbol{\mathrm{sin}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)}{\:\sqrt{\mathrm{2}}\boldsymbol{{a}}^{\mathrm{3}} \boldsymbol{\mathrm{cos}{h}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)−\boldsymbol{\mathrm{cos}}\left(\sqrt{\mathrm{2}}\boldsymbol{{a}\pi}\right)} \\ $$$$ \\ $$ Terms…
Question Number 211563 by MrGaster last updated on 13/Sep/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}.\boldsymbol{{x}}^{\mathrm{3}} −\mathrm{3}\boldsymbol{{x}}−\mathrm{2}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}.\boldsymbol{{e}}^{\boldsymbol{{x}}} +\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{4}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$ Answered by…
Question Number 211562 by MrGaster last updated on 13/Sep/24 $$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{1}}\\{\boldsymbol{{x}}^{\mathrm{3}} −\boldsymbol{\mathrm{y}}=\mathrm{0}}\end{cases} \\ $$ Answered by Rasheed.Sindhi last updated on 13/Sep/24 $$\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\boldsymbol{{x}}^{\mathrm{2}}…