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lim-x-0-1-2x-1-3x-1-3-x-2-




Question Number 167028 by cortano1 last updated on 04/Mar/22
      lim_(x→0)  (((√(1+2x))−((1+3x))^(1/3) )/x^2 ) =?
$$\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+\mathrm{2x}}−\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{3x}}}{\mathrm{x}^{\mathrm{2}} }\:=? \\ $$
Answered by qaz last updated on 05/Mar/22
lim_(x→0) (((√(1+2x))−((1+3x))^(1/3) )/x^2 )  =lim_(x→0) (((1+x−(1/2)x^2 )−(1+x−x^2 )+O(x^3 ))/x^2 )  =(1/2)
$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{1}+\mathrm{2x}}−\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{3x}}}{\mathrm{x}^{\mathrm{2}} } \\ $$$$=\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{1}+\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}^{\mathrm{2}} \right)−\left(\mathrm{1}+\mathrm{x}−\mathrm{x}^{\mathrm{2}} \right)+\mathcal{O}\left(\mathrm{x}^{\mathrm{3}} \right)}{\mathrm{x}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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