aghaee

\(\lim \limits_{x\to 0} { \left( \frac{ \sin{(x)} } {x} \right)^{ \cot{(x)} } }= \ ?\\ \)

Part 1:

We need

\(\boxed{~ \begin{array}{rcl} \cot{(x)} &=& \frac{1}{x} -\frac13 x -\frac{1}{45} x^3 -\frac{2}{945} x^5 -\frac{1}{4725} x^7 -\cdots \\ \ln{(1+x)} &=& x -\frac{x^2}{2} +\frac{x^3}{3} -\frac{x^4}{4} +-\cdots \\ \hline \sin{(x)} &=& x -\frac{x^3}{3!} +\frac{x^5}{5!} -\frac{x^7}{7!} +\frac{x^9}{9!} -+\cdots \\ \sin{(x)}-x &=& -\frac{x^3}{3!} +\frac{x^5}{5!} -\frac{x^7}{7!} +\frac{x^9}{9!} -+\cdots \\ \frac{\sin{(x)}-x}{x} &=& -\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \\ \hline \frac{\sin{(x)}}{x} &=& 1+ \left( \frac{\sin{(x)}}{x} - 1 \right) = 1+ \frac{\sin{(x)}-x}{x} \end{array} ~}\)

\(\boxed{~ \begin{array}{rcl} \ln{(1+z)} &=& z -\frac{z^2}{2} +\frac{z^3}{3} -\frac{z^4}{4} +-\cdots \\ z &=& \frac{\sin{(x)}-x}{x} = -\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \\ \ln{ ( 1+ \frac{ \sin{(x)}-x } {x} ) } &=& \left( -\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \right)^1 \\ &&-\frac12 \left(-\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \right)^2\\ &&+\frac13 \left(-\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \right)^3\\ &&-\frac14 \left(-\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \right)^4 \end{array} ~}\)

continued...

Heureka could you do it please ? 

This is what I think ://

lim_(x->0)(sinx/x)^cotx

=lim_(x->0)(1)^cotx

=1

Just to give a visual confirmation of Melody's answer:

.

Part 2:

\(\begin{array}{rcl} \lim \limits_{x\to 0} { \left( \frac{ \sin{(x)} } {x} \right)^{ \cot{(x)} } }=\ ?\\ \left( \frac{ \sin{(x)} } {x} \right)^{ \cot{(x)} } = e^{ \ln{ \left( \frac{ \sin{(x)} } {x} \right) \cdot \cot{(x)} } } = e^{ \ln{ \left( 1+ \frac{\sin{(x)}-x}{x} \right) } \cdot \cot{(x)} }\\\\ \ln{ \left( 1+ \frac{\sin{(x)}-x}{x} \right) } \cdot \cot{(x)} = \left[ \left( -\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \right)^1 \\ -\frac12 \left(-\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \right)^2\\ +\frac13 \left(-\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \right)^3\\ -\frac14 \left(-\frac{x^2}{3!} +\frac{x^4}{5!} -\frac{x^6}{7!} +\frac{x^8}{9!} -+\cdots \right)^4 \right] \cdot \\ \left( \frac{1}{x} -\frac13 x -\frac{1}{45} x^3 -\frac{2}{945} x^5 -\frac{1}{4725} x^7 -\cdots \right) \end{array}\)

\(\begin{array}{rcl} \ln{ \left( 1+ \frac{\sin{(x)}-x}{x} \right) } \cdot \cot{(x)} = \left[ \left( -\frac{x^2}{6} +\frac{x^4}{120} -\frac{x^6}{5040} +\frac{x^8}{40320} -+\cdots \right)\\ -\frac12 \left(\frac{x^4}{36} -\frac{x^6}{360} +\frac{41 x^8}{302400} -+\cdots \right)\\ +\frac13 \left(-\frac{x^6}{216} +\frac{x^8}{1440} -+\cdots \right)\\ -\frac14 \left(\frac{x^8}{1296} -+\cdots \right) \right] \cdot \\ \left( \frac{1}{x} -\frac13 x -\frac{1}{45} x^3 -\frac{2}{945} x^5 -\frac{1}{4725} x^7 -\cdots \right) \end{array} \)

until \(\mathbf{x^7}\):

\(\begin{array}{llcl} \ln{ \left( 1+ \frac{\sin{(x)}-x}{x} \right) } \cdot \cot{(x)} = \\ \mathbf{ \times (\frac{1}{x} ): } & -\frac{x}{6} +\frac{x^3}{120} -\frac{x^5}{5040} +\frac{x^7}{40320} -+\cdots \\ &-\frac{x^3}{72} +\frac{x^5}{720} -\frac{41x^7}{604800} +-\cdots\\ &-\frac{x^5}{648} +\frac{x^7}{4320} -+\cdots -\frac{x^7}{5184} +-\cdots\\ \mathbf{ \times (-\frac{1x}{3} ): } & +\frac{x^3}{18} -\frac{x^5}{360} +\frac{x^7}{15120} -+\cdots +\frac{x^5}{216} -\frac{x^7}{2160} +-\cdots +\frac{x^7}{1944} -+\cdots\\ \mathbf{ \times (-\frac{1x^3}{45} ): } & +\frac{x^5}{270} -\frac{x^7}{5400} -+\cdots +\frac{x^7}{3240} -+\cdots\\ \mathbf{ \times (-\frac{2x^5}{945} ): } & +\frac{x^7}{2835} -+\cdots \end{array}\)

\(\begin{array}{llcl} \ln{ \left( 1+ \frac{\sin{(x)}-x}{x} \right) } \cdot \cot{(x)} = \\ -\frac{x}{6}\\ +x^3(\frac{1}{120}-\frac{1}{72}+\frac{1}{18})\\ +x^5(-\frac{1}{5040}+\frac{1}{720}-\frac{1}{648}-\frac{1}{360}+\frac{1}{216}+\frac{1}{270})\\ +x^7(\frac{1}{40320}-\frac{41}{604800}+\frac{1}{4320}-\frac{1}{5184}+\frac{1}{15120}-\frac{1}{2160}+\frac{1}{1944}-\frac{1}{5400}+\frac{1}{3240}+\frac{1}{2835})\\ +\cdots\\\\ \boxed{~\ln{ \left( 1+ \frac{\sin{(x)}-x}{x} \right) } \cdot \cot{(x)} = -\frac{x}{6}+\frac{x^3}{20} +\frac{59x^5}{11340}+\frac{401x^7}{680400} +\cdots ~}\\\\ \lim \limits_{x\to 0} { \left( \frac{ \sin{(x)} } {x} \right)^{ \cot{(x)} } }\\ = \lim \limits_{x\to 0} { e^{ \ln{ \left( \frac{ \sin{(x)} } {x} \right) \cdot \cot{(x)} } } }\\ = \lim \limits_{x\to 0} { e^{ \ln{ \left( 1+ \frac{\sin{(x)}-x}{x} \right) } \cdot \cot{(x)} } }\\ = \lim \limits_{x\to 0} { e^{-\frac{x}{6}+\frac{x^3}{20} +\frac{59x^5}{11340}+\frac{401x^7}{680400} +\cdots } } =e^0\\ \boxed{~ \lim \limits_{x\to 0} { \left( \frac{ \sin{(x)} } {x} \right)^{ \cot{(x)} } } =1 ~} \end{array}\)

ready.