Question Number 194482 by horsebrand11 last updated on 08/Jul/23
$$\:\:\:\:\:\begin{array}{|c|}{\:\cancel{\underline{\underbrace{\Subset}}}}\\\hline\end{array} \\ $$
Answered by cortano12 last updated on 08/Jul/23
![L=lim_(x→0) [(1/x^2 ) ((2/(cos x)) + cos x−3)] L=lim_(x→0) (((cos^2 x−3cos x+2)/(x^2 cos x))) L= lim_(x→0) ((((cos x−2)(cos x−1))/(x^2 cos x))) L= lim_(x→0) (((cos x−2)/(cos x))).lim_(x→0) (((−sin^2 x)/(x^2 (cos x+1)))) L=−1×(−(1/2))= determinant (((1/2)))](https://www.tinkutara.com/question/Q194487.png)
$$\:\:\:\:\mathrm{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left[\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\:\left(\frac{\mathrm{2}}{\mathrm{cos}\:\mathrm{x}}\:+\:\mathrm{cos}\:\mathrm{x}−\mathrm{3}\right)\right] \\ $$$$\:\:\:\mathrm{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}−\mathrm{3cos}\:\mathrm{x}+\mathrm{2}}{\mathrm{x}^{\mathrm{2}} \mathrm{cos}\:\mathrm{x}}\right) \\ $$$$\:\:\mathrm{L}=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\left(\mathrm{cos}\:\mathrm{x}−\mathrm{2}\right)\left(\mathrm{cos}\:\mathrm{x}−\mathrm{1}\right)}{\mathrm{x}^{\mathrm{2}} \:\mathrm{cos}\:\mathrm{x}}\right) \\ $$$$\:\:\mathrm{L}=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{cos}\:\mathrm{x}−\mathrm{2}}{\mathrm{cos}\:\mathrm{x}}\right).\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{−\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}{\mathrm{x}^{\mathrm{2}} \left(\mathrm{cos}\:\mathrm{x}+\mathrm{1}\right)}\right) \\ $$$$\:\:\mathrm{L}=−\mathrm{1}×\left(−\frac{\mathrm{1}}{\mathrm{2}}\right)=\:\begin{array}{|c|}{\frac{\mathrm{1}}{\mathrm{2}}}\\\hline\end{array} \\ $$