Question Number 182560 by universe last updated on 11/Dec/22
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:−\:\mathrm{cot}^{\mathrm{2}} {x}\:=\:\:? \\ $$
Answered by cortano1 last updated on 11/Dec/22
$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{x}−\mathrm{x}}{\mathrm{x}^{\mathrm{3}} }\:.\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\mathrm{x}+\mathrm{x}}{\mathrm{x}} \\ $$$$\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:×\:\mathrm{2}=\frac{\mathrm{2}}{\mathrm{3}} \\ $$