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Author: Tinku Tara

lim-x-0-1-ln-1-x-1-ln-x-1-x-2-

Question Number 199843 by universe last updated on 10/Nov/23 $$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{ln}\left(\mathrm{1}+{x}\right)\:}−\frac{\mathrm{1}}{\mathrm{ln}\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\right)}\right)\:=\:?? \\ $$ Answered by witcher3 last updated on 10/Nov/23 $$=\frac{\mathrm{ln}\left(\mathrm{x}+\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\right)−\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{ln}\left(\left(\mathrm{1}+\mathrm{x}\right)\mathrm{ln}\left(\mathrm{x}+\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\right)\right.}= \\…

Question-199837

Question Number 199837 by Calculusboy last updated on 10/Nov/23 Answered by AST last updated on 10/Nov/23 $$\sqrt{{a}+\sqrt{{b}}}={p};\sqrt{{a}−\sqrt{{b}}}={q}\Rightarrow{p}^{\mathrm{2}} +{q}^{\mathrm{2}} =\mathrm{2}{a};{p}^{\mathrm{2}} {q}^{\mathrm{2}} ={a}^{\mathrm{2}} −{b} \\ $$$$\Rightarrow{pq}=\sqrt{{a}^{\mathrm{2}} −{b}}\Rightarrow\left({p}+{q}\right)^{\mathrm{2}}…

Question-199838

Question Number 199838 by Calculusboy last updated on 10/Nov/23 Answered by Frix last updated on 10/Nov/23 $${y}=\sqrt[{{x}}]{{xy}} \\ $$$${y}={x}^{\frac{\mathrm{1}}{{x}−\mathrm{1}}} \\ $$$$\left({x}+\mathrm{5}\right)^{{x}} =\mathrm{23}{x}+\mathrm{1}+{x}^{\frac{\mathrm{1}}{{x}−\mathrm{1}}} \\ $$$$\mathrm{Trying}\:{x}=\mathrm{1},\:\mathrm{2},\:\mathrm{3} \\…

Let-C-be-the-circle-with-the-center-2-3-and-radius-5-a-show-that-P-5-7-lies-on-C-and-find-the-equation-of-the-tangent-at-P-b-show-that-the-line-3x-4y-31-0-is-a-tangent-to-C-

Question Number 199834 by Calculusboy last updated on 10/Nov/23 $$\boldsymbol{{Let}}\:\boldsymbol{{C}}\:\boldsymbol{{be}}\:\boldsymbol{{the}}\:\boldsymbol{{circle}}\:\boldsymbol{{with}}\:\boldsymbol{{the}}\:\boldsymbol{{center}}\:\left(\mathrm{2},\mathrm{3}\right)\:\boldsymbol{{and}}\:\boldsymbol{{radius}}\:\mathrm{5} \\ $$$$\left.\boldsymbol{{a}}\right)\:\boldsymbol{{show}}\:\boldsymbol{{that}}\:\boldsymbol{{P}}\left(\mathrm{5},\mathrm{7}\right)\:\boldsymbol{{lies}}\:\boldsymbol{{on}}\:\boldsymbol{{C}}\:\boldsymbol{{and}}\:\boldsymbol{{find}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{equation}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{tangent}}\:\boldsymbol{{at}}\:\boldsymbol{{P}} \\ $$$$\left.\boldsymbol{{b}}\right)\:\boldsymbol{{show}}\:\boldsymbol{{that}}\:\boldsymbol{{the}}\:\boldsymbol{{line}}\:\mathrm{3}\boldsymbol{{x}}−\mathrm{4}\boldsymbol{{y}}+\mathrm{31}=\mathrm{0}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{tangent}}\:\boldsymbol{{to}}\:\boldsymbol{{C}} \\ $$ Commented by Calculusboy last updated on 10/Nov/23…

Question-199893

Question Number 199893 by hardmath last updated on 10/Nov/23 Commented by mr W last updated on 10/Nov/23 $${do}\:{you}\:{know}\:{what}\:{the}\:{question}\:{is}? \\ $$$${if}\:{you}\:{don}'{t}\:{know},\:{then}\:{other}\:{people} \\ $$$${don}'{t}\:{know}\:{either}. \\ $$$${badly}\:{prepared}\:{questions}\:{don}'{t} \\…

Si-cos-x-sin-x-1-3-2-halle-el-valor-de-la-expresion-R-16-sin-6-x-cos-6-x-3-sec-2-x-csc-2-x-

Question Number 199830 by cortano12 last updated on 10/Nov/23 $$\:\:\mathrm{Si}\:\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}=\frac{\mathrm{1}+\sqrt{\mathrm{3}}}{\mathrm{2}}\:, \\ $$$$\:\mathrm{halle}\:\mathrm{el}\:\mathrm{valor}\:\mathrm{de}\:\mathrm{la}\:\mathrm{expresion}\: \\ $$$$\:\mathrm{R}=\:\mathrm{16}\left(\mathrm{sin}\:^{\mathrm{6}} \mathrm{x}+\mathrm{cos}\:^{\mathrm{6}} \mathrm{x}\right)+\mathrm{3}\left(\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}+\mathrm{csc}^{\mathrm{2}} \:\mathrm{x}\right) \\ $$ Answered by Frix last updated…