\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt[]{2x+1}-\sqrt[9]{9x+1}}{\sqrt[]{4x+1}-1}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\left(\sqrt[]{2x+1}-1\right)-\left(\sqrt[3]{9x+1}-1\right)}{\sqrt[]{4x+1}-1}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{2x}{\sqrt[]{2x+1}+1}-\dfrac{9x}{\sqrt[3]{\left(9x+1\right)^2}+\sqrt[3]{9x+1}+1}}{\dfrac{4x}{\sqrt[]{4x+1}+1}}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{2}{\sqrt[]{2x+1}+1}-\dfrac{9}{\sqrt[3]{\left(9x+1\right)^2}+\sqrt[3]{9x+1}+1}}{\dfrac{4}{\sqrt[]{4x+1}+1}}=\dfrac{1-3}{2}=-1\)
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