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Asked by eknathmote50 | 02 Jun, 2019, 14:26: PM
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L equals limit as x rightwards arrow 0 of fraction numerator 1 minus cos 3 x space cos 9 x space cos 27 x space.... space cos 3 to the power of n x over denominator 1 minus cos begin display style 1 third end style x space cos begin display style 1 over 3 squared end style x space cos begin display style 1 over 3 cubed end style x... cos begin display style 1 over 3 to the power of n end style x end fraction L e t space P equals limit as x rightwards arrow 0 of fraction numerator 1 minus cos 3 x space cos 9 x space cos 27 x space.... space cos 3 to the power of n x over denominator x squared end fraction space a n d space Q equals limit as x rightwards arrow 0 of fraction numerator 1 minus cos 1 third x space cos 1 over 3 squared x space cos 1 over 3 cubed x... cos 1 over 3 to the power of n x over denominator x squared end fraction P equals limit as x rightwards arrow 0 of fraction numerator negative begin display style fraction numerator d over denominator d x end fraction end style open parentheses cos 3 x space cos 9 x space cos 27 x space.... space cos 3 to the power of n x close parentheses over denominator 2 x end fraction L e t space y equals cos 3 x space cos 9 x space cos 27 x space.... space cos 3 to the power of n x rightwards double arrow ln y equals ln space cos 3 x plus space ln space cos 9 x plus space... space ln space cos 3 to the power of n x rightwards double arrow 1 over y fraction numerator d y over denominator d x end fraction equals 3 x space tan 3 x plus 9 x space tan 9 x plus... plus 3 to the power of n space tan 3 to the power of n x rightwards double arrow fraction numerator d y over denominator d x end fraction equals product from r equals 1 to n of space cos 3 to the power of r x open square brackets 3 x space tan 3 x plus 9 x space tan 9 x plus... plus 3 to the power of n space tan 3 to the power of n x close square brackets rightwards double arrow P equals limit as x rightwards arrow 0 of fraction numerator 3 space tan 3 x plus 3 squared space tan 3 squared x plus... plus 3 to the power of n space tan 3 to the power of n x over denominator 2 x end fraction equals fraction numerator 3 squared plus open parentheses 3 squared close parentheses squared plus... plus open parentheses 3 to the power of n close parentheses squared over denominator 2 end fraction rightwards double arrow P equals fraction numerator 3 squared open parentheses 3 to the power of 2 n end exponent minus 1 close parentheses over denominator open parentheses 3 squared minus 1 close parentheses 2 end fraction S i m i l a r l y comma space Q equals fraction numerator begin display style 1 over 3 squared end style open parentheses 1 minus begin display style 1 over 3 to the power of 2 n end exponent end style close parentheses over denominator open parentheses 1 minus 1 over 3 squared close parentheses 2 end fraction equals fraction numerator open parentheses 3 to the power of 2 n end exponent minus 1 close parentheses over denominator 3 to the power of 2 n end exponent open parentheses 3 squared minus 1 close parentheses 2 end fraction T h e r e f o r e comma limit as x rightwards arrow 0 of fraction numerator 1 minus cos 3 x space cos 9 x space cos 27 x space.... space cos 3 to the power of n x over denominator 1 minus cos begin display style 1 third end style x space cos begin display style 1 over 3 squared end style x space cos begin display style 1 over 3 cubed end style x... cos begin display style 1 over 3 to the power of n end style x end fraction equals P over Q equals fraction numerator 3 squared over denominator begin display style 1 over 3 to the power of 2 n end exponent end style end fraction equals 3 to the power of 2 plus 2 n end exponent equals 3 to the power of 10 space left parenthesis g i v e n right parenthesis rightwards double arrow 2 plus 2 n equals 10 rightwards double arrow n equals 4
Answered by Renu Varma | 06 Jun, 2019, 11:28: AM

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