Question Number 194891 by cortano12 last updated on 18/Jul/23
$$\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{3}^{+} } {\mathrm{lim}}\:\left(\frac{\sqrt{{x}}−\sqrt{{x}−\mathrm{3}}−\sqrt{\mathrm{3}}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{9}}}\:\right)=? \\ $$
Answered by MM42 last updated on 18/Jul/23
$${lim}_{{x}\rightarrow\mathrm{3}} \:\left(\frac{\sqrt{{x}}−\sqrt{\mathrm{3}}}{\:\sqrt{{x}−\mathrm{3}}×\sqrt{{x}+\mathrm{3}}}\:−\:\frac{\sqrt{{x}−\mathrm{3}}}{\:\sqrt{{x}−\mathrm{3}}×\sqrt{{x}+\mathrm{3}}}\:\right) \\ $$$$={lim}_{{x}\rightarrow\mathrm{3}} \:\left(\frac{\sqrt{{x}−\mathrm{3}}×\sqrt{{x}−\mathrm{3}}}{\:\left(\sqrt{{x}}+\sqrt{\mathrm{3}}\right)×\sqrt{{x}−\mathrm{3}}×\sqrt{{x}+\mathrm{3}}}\:−\:\frac{\sqrt{{x}−\mathrm{3}}}{\:\sqrt{{x}−\mathrm{3}}×\sqrt{{x}+\mathrm{3}}}\:\right) \\ $$$$={lim}_{{x}\rightarrow\mathrm{3}} \:\left(\frac{\sqrt{{x}−\mathrm{3}}}{\:\left(\sqrt{{x}}+\sqrt{\mathrm{3}}\right)×\sqrt{{x}+\mathrm{3}}}\:−\:\frac{\mathrm{1}}{\:\sqrt{{x}+\mathrm{3}}}\:\right) \\ $$$$=−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:\checkmark \\ $$$$ \\ $$
Commented by cortano12 last updated on 18/Jul/23
$$\:\:=\:−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}\:}\:\left(×\right) \\ $$$$\:\:=\:−\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}}\:\left(\checkmark\right) \\ $$
Commented by MM42 last updated on 18/Jul/23
$${ok} \\ $$$${it}\:{was}\:{my}\:{mistake} \\ $$$${x}\rightarrow\mathrm{3}\Rightarrow\:−\frac{\mathrm{1}}{\:\sqrt{{x}+\mathrm{3}}}=−\frac{\mathrm{1}}{\:\sqrt{\mathrm{6}}} \\ $$