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Question Number 140471 by EDWIN88 last updated on 08/May/21

$$\mathrm{If}\underset{x\to 0}{\lim }{(\cos \mathrm{x}+\mathrm{a}\sin \mathrm{bx})}^{\frac{1}{\mathrm{x}}}={\mathrm{e}}^2$$$$\{\begin{array}{l}\mathrm{a}=?\\ \mathrm{b}=?\end{array}$$

Answered by benjo_mathlover last updated on 08/May/21

$$\underset{x\to 0}{\lim }{(\cos \mathrm{x}+\mathrm{a}\sin \mathrm{bx})}^{\frac{1}{\mathrm{x}}}=$$$${\mathrm{e}}^{\underset{x\to 0}{\lim }(\cos \mathrm{x}+\mathrm{a}\sin \mathrm{bx}-1).\frac{1}{\mathrm{x}}}=$$$${\mathrm{e}}^{\underset{x\to 0}{\lim }\left(\frac{\cos \mathrm{x}-1+\mathrm{a}\sin \mathrm{bx}}{\mathrm{x}}\right)}=$$$${\mathrm{e}}^{\underset{x\to 0}{\lim }\left(\frac{-\sin \mathrm{x}+\mathrm{ab}\cos \mathrm{x}}{1}\right)}={\mathrm{e}}^{\mathrm{ab}}={\mathrm{e}}^2$$$$\Rightarrow \mathrm{ab}=2\to \{\begin{array}{l}\mathrm{a}=\mathrm{k}\\ \mathrm{b}=\frac{2}{\mathrm{k}}\end{array}$$$$$$

Answered by mathmax by abdo last updated on 09/May/21

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)={(\mathrm{cosx}+\mathrm{asinbx})}^{\frac{1}{\mathrm{x}}}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\frac{1}{\mathrm{x}}\log (\mathrm{cosx}+\mathrm{asinx})}$$$$\mathrm{cosx}\sim 1-\frac{{\mathrm{x}}^2}{2}\mathrm{and}\mathrm{sinbx}\sim \mathrm{bx}\Rightarrow \mathrm{cosx}+\mathrm{asin}\left(\mathrm{bx}\right)\sim 1-\frac{{\mathrm{x}}^2}{2}+\mathrm{abx}\Rightarrow$$$$\log (\mathrm{cosx}+\mathrm{asin}\left(\mathrm{bx}\right))\sim \log (1+\mathrm{abx}-\frac{{\mathrm{x}}^2}{2})\sim \mathrm{abx}-\frac{{\mathrm{x}}^2}{2}\Rightarrow$$$$\frac{1}{\mathrm{x}}\log (\mathrm{cosx}+\mathrm{asin}\left(\mathrm{bx}\right))\sim \mathrm{ab}-\frac{\mathrm{x}}{2}\to \mathrm{ab}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)\to {\mathrm{e}}^{\mathrm{ab}}={\mathrm{e}}^2\Rightarrow \mathrm{ab}=2$$

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