For n = 1
\(\lim\limits_{x\to 0} \cfrac{1 - cos 3x}{1 - cos \frac x3} \) \(\left(\frac 00 - case\right)\)
\(= \lim\limits_{x \to 0} \cfrac {3sin \,3x}{\frac 13 sin\frac x3}\)
\(= \lim\limits_{x\to 0} \cfrac{3^2 \frac {sin 3x}{3x}. \,3x}{\frac{sin \frac x3}{\frac x3}.\,\frac x3}\)
\(= \lim\limits_{x \to 0} 3^4 \;\frac {sin 3x}{3x}\; \frac {\frac x3}{sin\frac x3}\)
\( = 3^4 \ne 3^{10}\)
For n = 2
\(\lim\limits_{x\to0} \cfrac {1 - cos3x \,cos 9x}{1 - cos \frac x3\, cos \frac x9}\)
\(= \lim\limits_{x \to 0} \cfrac{9cos 3x \,sin 9x + 3sin3x \,cos 9x}{\frac 19 cos \frac x3\, sin\frac x9 + \frac 13 sin\frac x3\,cos\frac x9}\)
\(= \cfrac {81 + 9}{\frac 1{81} + \frac 19}\)
\(= \cfrac {90}{\frac {9+1}{81}} \)
\(= 90 \times \frac{81}{10} \)
\(= 729 \)
\(= 3^6\)
For n = 3
\(\lim\limits_{x\to 0} \cfrac {1 - cos 3x \,cos 9x\,cos 27 x}{1 - cos\frac x3 \, cos \frac x9\, cos \frac x{27}}\)
\(= \lim\limits_{x\to 0} \cfrac {3\,sin 3x \, cos 9x\, cos27x + 9\,cos3x \, sin9x\, cos27x + 27\,cos3x \, cos 9x \, sin 27x \, }{\frac 13 \,sin \frac x3 \, cos \frac x9 \, cos \frac x{27} + \frac 19\, cos \frac x3 \, sin \frac x9 \, cos \frac x{27}+ \frac 1{27}\,cos \frac x3\, cos \frac x9 \, sin \frac x{27}}\)
\(=\cfrac {9 + 81 + 729}{\frac 1{9} + \frac 1{81} + \frac 1{729}}\)
\(= \cfrac {819}{\frac{81 + 9 + 1}{729}}\)
\(= 819 \times \frac{729}{91} \)
\(= 9 \times 729 \)
\(= 3^8\)
For n = 4
\(\lim\limits_{x\to 0} \cfrac {1 - cos 3x \, cos 9x\, cos 27x\, cos 81x}{1 - cos \frac x3 \, cos \frac x9 \, cos\frac x{27} \, cos\frac x{81}}\)
\(= \cfrac{3^2 + 9^2 + 27^2 + 81^2}{\left(\frac13\right)^2 + \left(\frac19\right)^2 + \left(\frac1{27}\right)^2 + \left(\frac1{81}\right)^2}\)
\(= \cfrac {9 + 81 + 729 + 6561}{ \frac 19 + \frac 1{81} + \frac 1{729} + \frac 1{6561}}\)
\(= \cfrac {7380}{\frac{729 + 81 + 9 + 1}{6561}}\)
\(= 7380 \times \frac {6561}{820}\)
\(= 9 \times 6561\)
\(= 3^2 \times 3^8\)
\(= 3^{10}\)
Hence, the value of n is 4.
