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Evaluate the limit as x-> 0?

0 votes

(sin(3x)sin(5x))/x^2

limits

asked May 27, 2013 in PRECALCULUS by anonymous Apprentice

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1 Answer

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$italic lim with x rightwards arrow 0 below fraction numerator italic sin 3 x cross times italic sin 5 x over denominator x to the power of 2 end fraction equal italic lim with x rightwards arrow 0 below fraction numerator italic sin 3 x cross times italic sin 5 x over denominator x cross times x end fraction$

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equal italic lim with x rightwards arrow 0 below fraction numerator italic sin 3 x over denominator x end fraction cross times fraction numerator italic sin 5 x over denominator x end fraction$

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equal italic lim with x rightwards arrow 0 below fraction numerator italic sin 3 x over denominator x end fraction cross times italic lim with x rightwards arrow 0 below fraction numerator italic sin 5 x over denominator x end fraction$ (Apply limits product rule)

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equal italic lim with x rightwards arrow 0 below fraction numerator italic sin 3 x over denominator 3 x end fraction open parentheses 3 close parentheses cross times italic lim with x rightwards arrow 0 below fraction numerator italic sin 5 x over denominator 5 x end fraction open parentheses 5 close parentheses$

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equal 3 italic lim with 3 x rightwards arrow 0 below fraction numerator italic sin 3 x over denominator 3 x end fraction cross times 5 italic lim with 5 x rightwards arrow 0 below fraction numerator italic sin 5 x over denominator 5 x end fraction$ (x stands to 0 then 3x stands to 0)

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equal 3 open parentheses 1 close parentheses cross times 5 open parentheses 1 close parentheses

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equal 3 cross times 5

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equal 15.