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Question Number 131745 by liberty last updated on 08/Feb/21

Commented by guyyy last updated on 18/Feb/21

Commented by guyyy last updated on 18/Feb/21

Commented by guyyy last updated on 18/Feb/21

Commented by guyyy last updated on 18/Feb/21

Answered by bemath last updated on 08/Feb/21

$$\left(2\right)\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }{\left[{(1+\frac{1}{{\mathrm{n}}^2})}^{{\mathrm{n}}^2}\right]}^{\frac{4\left(\sin \frac{\pi }{2\mathrm{n}}\right)}{\frac{\pi }{2\mathrm{n}}}}={\mathrm{e}}^{\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\left(\frac{4\sin \frac{\pi }{2\mathrm{n}}}{\frac{\pi }{2\mathrm{n}}}\right)}={\mathrm{e}}^4$$

Answered by bemath last updated on 08/Feb/21

$$\left(1\right)\mathrm{since}{5\mathrm{x}}^2-18\mathrm{is}\mathrm{a}\mathrm{quadratic}\mathrm{polynomial}$$$$\mathrm{and}\mathrm{P}\left(\mathrm{x}\right)+\mathrm{P}\left(2\mathrm{x}\right)={5\mathrm{x}}^2-18\mathrm{it}\mathrm{follows}\mathrm{that}\mathrm{P}\left(\mathrm{x}\right)$$$$\mathrm{must}\mathrm{be}\mathrm{a}\mathrm{quadratic}\mathrm{polynomial}.$$$$\mathrm{let}\mathrm{P}\left(\mathrm{x}\right)={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$$$$\mathrm{P}\left(2\mathrm{x}\right)={4\mathrm{ax}}^2+2\mathrm{bx}+\mathrm{c}$$$$\Rightarrow {5\mathrm{ax}}^2+3\mathrm{bx}+2\mathrm{c}\equiv {5\mathrm{x}}^2-18\to \{\begin{array}{l}\mathrm{a}=1,\mathrm{b}=0\\ \mathrm{c}=-9\end{array}$$$$\underset{x\to 3}{\lim }\left(\frac{\mathrm{P}\left(\mathrm{x}\right)}{\mathrm{x}-3}\right)=\underset{x\to 3}{\lim }\left(\frac{{\mathrm{x}}^2-9}{\mathrm{x}-3}\right)=6$$

Answered by bemath last updated on 08/Feb/21

$$\left(3\right)\mathrm{let}\alpha ={\sin }^{-1}\sqrt{\frac{7}{8}}\mathrm{then}$$$$\underset{x\to \alpha }{\lim }\frac{\tan {}^2\mathrm{x}-7}{8\sin {}^2\mathrm{x}-7(\sin {}^2\mathrm{x}+\cos {}^2\mathrm{x})}=$$$$\underset{x\to \alpha }{\lim }\frac{\tan {}^2\mathrm{x}-7}{\sin {}^2\mathrm{x}-7\cos {}^2\mathrm{x}}=\underset{x\to \alpha }{\lim }\frac{1}{\cos {}^2\mathrm{x}}=\frac{1}{\cos {}^2\alpha }$$$$=8$$$$$$

Commented by guyyy last updated on 13/Feb/21

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