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Maths-

General

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Question

$Lt subscript x not stretchy rightwards arrow 0 end subscript invisible function application fraction numerator 1 minus cos invisible function application 2 m x over denominator sin squared invisible function application n x end fraction left parenthesis m comma n element of Z right parenthesis$

$\frac{2 m^{2}}{n^{2}}$
$\frac{m^{2}}{n^{2}}$
$\frac{n^{2}}{m^{2}}$
$\frac{2 n^{2}}{m^{2}}$

The correct answer is: $fraction numerator 2 m squared over denominator n squared end fraction$

$limit as x rightwards arrow 0 of open parentheses fraction numerator 1 minus cos 2 m x over denominator sin squared n x end fraction close parentheses equals limit as x rightwards arrow 0 of open parentheses fraction numerator 2 sin squared m x over denominator sin squared n x end fraction close parentheses equals limit as x rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator 2 sin squared m x over denominator x squared end fraction end style over denominator begin display style fraction numerator sin squared n x over denominator x squared end fraction end style end fraction close parentheses equals 2 limit as x rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator sin squared m x over denominator m squared x squared end fraction cross times m squared end style over denominator begin display style fraction numerator sin squared n x over denominator n squared x squared end fraction cross times n squared end style end fraction close parentheses equals fraction numerator 2 m squared over denominator n squared end fraction limit as x rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator sin squared m x over denominator m squared x squared end fraction end style over denominator begin display style fraction numerator sin squared n x over denominator n squared x squared end fraction end style end fraction close parentheses equals fraction numerator 2 m squared over denominator n squared end fraction cross times 1 equals fraction numerator 2 m squared over denominator n squared end fraction$

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