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  • Evaluate  <img alt="\mathrm{\lim _{x \rightarrow 0} \frac{\left(e^x-1\right)(\ln (1+\sin 3 x)}{\ln (1-\sin x)}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20

Evaluate  \mathrm{\lim _{x \rightarrow 0} \frac{\left(e^x-1\right)(\ln (1+\sin 3 x)}{\ln (1-\sin x)}}

Option: 1

0

 


Option: 2

1


Option: 3

2


Option: 4

-1


Answers (1)

best_answer

A necessary skill and prerequisite to solve limit is the knowledge of the standard limits; in this case we are referring to \mathrm{(as\; x \rightarrow 0 )}

\\\mathrm{\frac{e^{2-1}}{x}=1}\\ \\\mathrm{\frac{\sin x}{x}=1}\\ \\\mathrm{\frac{\log (1+x)}{x}=1}

using these results we have that

                        \begin{gathered} \mathrm{\frac{\left(e^x-1\right)(\ln (1+\sin (3 x)))}{\ln (1-\sin x)}= }\\ \\=\mathrm{\frac{e^x-1}{x} \cdot \frac{\ln (1+\sin (3 x))}{\sin (3 x)} \cdot \frac{-\sin x}{\ln (1-\sin x)} \cdot \frac{\sin (3 x)}{3 x} \cdot \frac{x}{-\sin x} \cdot 3 x \rightarrow} \\ \\\rightarrow 1 \cdot 1 \cdot 1 \cdot 1 \cdot(-1) \cdot 0=0 \end{gathered}

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