Question Number 155120 by mnjuly1970 last updated on 25/Sep/21
$$ \\ $$$$\:\:\:{lim}_{\:{x}\:\rightarrow\mathrm{0}} \left(\frac{\mathrm{1}}{{x}^{\:\mathrm{2}} }\:−\:{cot}^{\:\mathrm{2}} \left({x}\right)\right)=? \\ $$$$ \\ $$
Answered by john_santu last updated on 25/Sep/21
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }−\mathrm{cot}\:^{\mathrm{2}} {x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{tan}\:^{\mathrm{2}} {x}}\right) \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{tan}\:{x}+{x}}{\mathrm{tan}\:{x}}\right)×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{tan}\:{x}−{x}}{{x}^{\mathrm{2}} \:\mathrm{tan}\:{x}}\right) \\ $$$$=\mathrm{2}\:×\frac{\mathrm{1}}{\mathrm{3}}=\frac{\mathrm{2}}{\mathrm{3}} \\ $$
Commented by mnjuly1970 last updated on 25/Sep/21
$${grateful}… \\ $$