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Question-190765




Question Number 190765 by mustafazaheen last updated on 10/Apr/23
Answered by qaz last updated on 11/Apr/23
lim_(x→0) ((−e^2 +(1+2x)^(1/x) )/(sin x))=lim_(x→0) ((−e^2 +e^((1/x)ln(1+2x)) )/x)  =lim_(x→0) ((−e^2 +e^(2−2x+o(x)) )/x)=lim_(x→0) ((−e^2 +e^2 (1−2x+o(x)))/x)=−2e^2
$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{−{e}^{\mathrm{2}} +\left(\mathrm{1}+\mathrm{2}{x}\right)^{\frac{\mathrm{1}}{{x}}} }{\mathrm{sin}\:{x}}=\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{−{e}^{\mathrm{2}} +{e}^{\frac{\mathrm{1}}{{x}}{ln}\left(\mathrm{1}+\mathrm{2}{x}\right)} }{{x}} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{−{e}^{\mathrm{2}} +{e}^{\mathrm{2}−\mathrm{2}{x}+{o}\left({x}\right)} }{{x}}=\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{−{e}^{\mathrm{2}} +{e}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{2}{x}+{o}\left({x}\right)\right)}{{x}}=−\mathrm{2}{e}^{\mathrm{2}} \\ $$
Commented by mustafazaheen last updated on 12/Apr/23
Thanks Sir  o(x)=?
$$\mathrm{Thanks}\:\mathrm{Sir} \\ $$$$\mathrm{o}\left(\mathrm{x}\right)=? \\ $$

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