Question Number 147169 by mathdanisur last updated on 18/Jul/21 | ||
$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\left(\frac{\mathrm{3}^{\boldsymbol{{x}}} \:+\:\mathrm{5}^{\boldsymbol{{x}}} \:+\:\mathrm{6}^{\boldsymbol{{x}}} }{\mathrm{3}}\right)^{\frac{\mathrm{1}}{\boldsymbol{{x}}}} =\:? \\ $$ | ||
Commented by gsk2684 last updated on 18/Jul/21 | ||
$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\left(\frac{\mathrm{3}^{\boldsymbol{{x}}} \:+\:\mathrm{5}^{\boldsymbol{{x}}} \:+\:\mathrm{6}^{\boldsymbol{{x}}} }{\mathrm{3}}\right)^{\frac{\mathrm{1}}{\boldsymbol{{x}}}} =\:? \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{3}×\mathrm{5}×\mathrm{6}} \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{90}} \\ $$ | ||
Commented by mathdanisur last updated on 19/Jul/21 | ||
$${Thankyou}\:{Ser} \\ $$$${How}\:{did}\:{you}\:{get}\:{the}\:{answer}\:{is}\:{there} \\ $$$${a}\:{formula}? \\ $$ | ||
Commented by gsk2684 last updated on 21/Jul/21 | ||
$${yes}. \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{a}_{\mathrm{1}} ^{{x}} +{a}_{\mathrm{2}} ^{{x}} +...+{a}_{{n}} ^{{x}} }{{n}}\right)^{\frac{\mathrm{1}}{{x}}} \\ $$$${e}^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\left(\frac{{a}_{\mathrm{1}} ^{{x}} +{a}_{\mathrm{2}} ^{{x}} +...+{a}_{{n}} ^{{x}} }{{n}}−\mathrm{1}\right)} \\ $$$${e}^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\left(\frac{{a}_{\mathrm{1}} ^{{x}} +{a}_{\mathrm{2}} ^{{x}} +...+{a}_{{n}} ^{{x}} −{n}}{{n}}\right)} \\ $$$${e}^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\left(\frac{\left({a}_{\mathrm{1}} ^{{x}} −\mathrm{1}\right)+\left({a}_{\mathrm{2}} ^{{x}} −\mathrm{1}\right)+...+\left({a}_{{n}} ^{{x}} −\mathrm{1}\right)}{{x}}\right)} \\ $$$${e}^{\frac{\mathrm{1}}{{n}}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{a}_{\mathrm{1}} ^{{x}} −\mathrm{1}}{{x}}+\frac{{a}_{\mathrm{2}} ^{{x}} −\mathrm{1}}{{x}}+...+\frac{{a}_{{n}} ^{{x}} −\mathrm{1}}{{x}}\right)} \\ $$$${e}^{\frac{\mathrm{1}}{{n}}\left(\mathrm{ln}\:{a}_{\mathrm{1}} +\mathrm{ln}\:{a}_{\mathrm{2}} +...+\mathrm{ln}\:{a}_{{n}} \right)} \\ $$$${e}^{\frac{\mathrm{1}}{{n}}\mathrm{ln}\:\left({a}_{\mathrm{1}} \:{a}_{\mathrm{2}} \:...\:{a}_{{n}} \right)} \\ $$$${e}^{\mathrm{ln}\:\left({a}_{\mathrm{1}} \:{a}_{\mathrm{2}} \:...\:{a}_{{n}} \right)^{\frac{\mathrm{1}}{{n}}} } \\ $$$$\left({a}_{\mathrm{1}} {a}_{\mathrm{2}} ...{a}_{{n}} \right)^{\frac{\mathrm{1}}{{n}}} \\ $$$$\sqrt[{{n}}]{{a}_{\mathrm{1}} {a}_{\mathrm{2}} ...{a}_{{n}} } \\ $$ | ||
Commented by mathdanisur last updated on 21/Jul/21 | ||
$${Thank}\:{you}\:{Ser},\:{cool} \\ $$ | ||
Answered by mathmax by abdo last updated on 18/Jul/21 | ||
$$\mathrm{f}\left(\mathrm{x}\right)=\left(\frac{\mathrm{3}^{\mathrm{x}} \:+\mathrm{5}^{\mathrm{x}} \:+\mathrm{6}^{\mathrm{x}} }{\mathrm{3}}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} \:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)=\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}\mathrm{log}\left(\frac{\mathrm{3}^{\mathrm{x}\:} +\mathrm{5}^{\mathrm{x}} \:+\mathrm{6}^{\mathrm{x}} }{\mathrm{3}}\right)} \\ $$$$\mathrm{3}^{\mathrm{x}} \:=\mathrm{e}^{\mathrm{xlog3}} \:\sim\mathrm{1}+\mathrm{xlog3} \\ $$$$\mathrm{5}^{\mathrm{x}} \:=\mathrm{e}^{\mathrm{xlog5}} \:\sim\mathrm{1}+\mathrm{xlog5} \\ $$$$\mathrm{6}^{\mathrm{x}} \:=\mathrm{e}^{\mathrm{xlog6}} \:\sim\mathrm{1}+\mathrm{xlog6}\:\Rightarrow\frac{\mathrm{3}^{\mathrm{x}} \:+\mathrm{5}^{\mathrm{x}} \:+\mathrm{6}^{\mathrm{x}} }{\mathrm{3}}\sim\frac{\mathrm{3}+\mathrm{x}\left(\mathrm{log3}+\mathrm{log5}+\mathrm{log6}\right)}{\mathrm{3}} \\ $$$$=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{log}\left(\mathrm{90}\right)\mathrm{x}\:=\mathrm{1}+\frac{\mathrm{log}\left(\mathrm{90}\right)}{\mathrm{3}}\mathrm{x}\:\Rightarrow\right. \\ $$$$\mathrm{log}\left(\frac{\mathrm{3}^{\mathrm{x}} \:+\mathrm{6}^{\mathrm{x}} \:+\mathrm{6}^{\mathrm{x}} }{\mathrm{3}}\right)\sim\mathrm{log}\left(\mathrm{1}+\frac{\mathrm{log}\left(\mathrm{90}\right)}{\mathrm{3}}\mathrm{x}\right)\sim\frac{\mathrm{log}\left(\mathrm{90}\right)}{\mathrm{3}}\mathrm{x}\:\Rightarrow \\ $$$$\frac{\mathrm{1}}{\mathrm{x}}\mathrm{log}\left(\frac{\mathrm{3}^{\mathrm{x}} +\mathrm{5}^{\mathrm{x}} \:+\mathrm{6}^{\mathrm{x}} }{\mathrm{3}}\right)\sim\frac{\mathrm{log}\left(\mathrm{90}\right)}{\mathrm{3}}\:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\sim\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{3}}\mathrm{log}\left(\mathrm{90}\right)} \\ $$$$=\mathrm{e}^{\left(\mathrm{90}^{\frac{\mathrm{1}}{\mathrm{3}}} \right)} =^{\mathrm{3}} \sqrt{\mathrm{90}}\:\Rightarrow\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\:\mathrm{f}\left(\mathrm{x}\right)=^{\mathrm{3}} \sqrt{\mathrm{90}} \\ $$$$ \\ $$ | ||
Commented by mathdanisur last updated on 19/Jul/21 | ||
$${thank}\:{you}\:{Ser} \\ $$ | ||