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02/12/2023
04/12/2023
$\displaystyle \begin{array}{{>{\displaystyle}l}}
\lim _{x\rightarrow 1}\sqrt{1+3x} -\sqrt[3]{1-9x} =4\\
( x-1)^{2} >0\ \forall x\neq 1\\
\Longrightarrow \lim _{x\rightarrow 1}\frac{\sqrt{1+3x} -\sqrt[3]{1-9x}}{( x-1)^{2}} =+\infty
\end{array}$
03/12/2023
$\displaystyle \begin{array}{{>{\displaystyle}l}}
\lim\limits _{x\rightarrow 1}\frac{\sqrt{1+3x} -\sqrt[3]{1-9x}}{( x-1)^{2}}\\
=\lim\limits _{x\rightarrow 1}\frac{\sqrt{1+3x} -\left(\frac{3x}{4} +\frac{5}{4}\right) +\sqrt[3]{1-9x} +\left(\frac{3x}{4} +\frac{5}{4}\right)}{( x-1)^{2}}\\
=\lim\limits _{x\rightarrow 1}\frac{\sqrt{1+3x} -\left(\frac{3x}{4} +\frac{5}{4}\right)}{( x-1)^{2}} +\lim _{x\rightarrow 1}\frac{\sqrt[3]{1-9x} +\left(\frac{3x}{4} +\frac{5}{4}\right)}{( x-1)^{2}}\\
=\lim\limits _{x\rightarrow 1}\frac{\left[\sqrt{1+3x} -\left(\frac{3x}{4} +\frac{5}{4}\right)\right]\left[\sqrt{1+3x} +\left(\frac{3x}{4} +\frac{5}{4}\right)\right]}{( x-1)^{2}\left[\sqrt{1+3x} +\left(\frac{3x}{4} +\frac{5}{4}\right)\right]}\\
+\lim _{x\rightarrow 1}\frac{\left[\sqrt[3]{1-9x} +\left(\frac{3x}{4} +\frac{5}{4}\right)\right]\left[\sqrt[3]{( 1-9x)^{2}} -\sqrt[3]{1-9x}\left(\frac{3x}{4} +\frac{5}{4}\right) +\left(\frac{3x}{4} +\frac{5}{4}\right)^{2}\right]}{( x-1)^{2}\left[\sqrt[3]{( 1-9x)^{2}} -\sqrt[3]{1-9x}\left(\frac{3x}{4} +\frac{5}{4}\right) +\left(\frac{3x}{4} +\frac{5}{4}\right)^{2}\right]}\\
=\lim\limits _{x\rightarrow 1}\frac{1+3x-\left(\frac{3x}{4} +\frac{5}{4}\right)^{2}}{( x-1)^{2}\left[\sqrt{1+3x} +\left(\frac{3x}{4} +\frac{5}{4}\right)\right]} +\lim\limits _{x\rightarrow 1}\frac{1-9x+\left(\frac{3x}{4} +\frac{5}{4}\right)^{3}}{( x-1)^{2}\left[\sqrt[3]{( 1-9x)^{2}} -\sqrt[3]{1-9x}\left(\frac{3x}{4} +\frac{5}{4}\right) +\left(\frac{3x}{4} +\frac{5}{4}\right)^{2}\right]}\\
=\lim\limits _{x\rightarrow 1}\frac{1+3x-\frac{1}{16}\left( 9x^{2} +30x+25\right)}{( x-1)^{2}\left[\sqrt{1+3x} +\left(\frac{3x}{4} +\frac{5}{4}\right)\right]} +\lim\limits _{x\rightarrow 1}\frac{1-9x+\frac{1}{64}\left( 27x^{3} +135x^{2} +225x+125\right)}{( x-1)^{2}\left[\sqrt[3]{( 1-9x)^{2}} -\sqrt[3]{1-9x}\left(\frac{3x}{4} +\frac{5}{4}\right) +\left(\frac{3x}{4} +\frac{5}{4}\right)^{2}\right]}\\
=\lim\limits _{x\rightarrow 1}\frac{\frac{-9}{16}( x-1)^{2}}{( x-1)^{2}\left[\sqrt{1+3x} +\left(\frac{3x}{4} +\frac{5}{4}\right)\right]} +\lim\limits _{x\rightarrow 1}\frac{\frac{27}{64}( x-1)^{2}( x+7)}{( x-1)^{2}\left[\sqrt[3]{( 1-9x)^{2}} -\sqrt[3]{1-9x}\left(\frac{3x}{4} +\frac{5}{4}\right) +\left(\frac{3x}{4} +\frac{5}{4}\right)^{2}\right]}\\
=\frac{-9}{16}\lim\limits _{x\rightarrow 1}\frac{1}{\sqrt{1+3x} +\left(\frac{3x}{4} +\frac{5}{4}\right)} +\frac{27}{64}\lim\limits _{x\rightarrow 1}\frac{\frac{27}{64}( x+7)}{\left[\sqrt[3]{( 1-9x)^{2}} -\sqrt[3]{1-9x}\left(\frac{3x}{4} +\frac{5}{4}\right) +\left(\frac{3x}{4} +\frac{5}{4}\right)^{2}\right]}\\
=\frac{-9}{16} .\frac{1}{2+2} +\frac{27}{64} .\frac{8}{( -2)^{2} +2.2+2^{2}} =\frac{9}{64}
\end{array}$
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