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Maths-

General

Easy

Question

$L subscript x not stretchy rightwards arrow 0 end subscript fraction numerator 1 minus cos invisible function application m x over denominator 1 minus cos invisible function application n x end fraction comma n not equal to 0$

$\frac{n^{2}}{m^{2}}$
$\frac{m^{2}}{n^{2}}$
$m^{2} n^{2}$
0

The correct answer is: $m squared n squared$

$limit as x rightwards arrow 0 of fraction numerator 1 minus cos m x over denominator 1 minus cos n x end fraction W e space k n o w space t h a t space cos 2 x equals 1 minus sin squared x over 2 comma equals limit as x rightwards arrow 0 of fraction numerator 1 minus open parentheses 1 minus sin squared begin display style fraction numerator m x over denominator 2 end fraction end style close parentheses over denominator 1 minus open parentheses 1 minus sin squared fraction numerator n x over denominator 2 end fraction close parentheses end fraction equals limit as x rightwards arrow 0 of open square brackets fraction numerator sin squared open parentheses begin display style fraction numerator m x over denominator 2 end fraction end style close parentheses over denominator sin squared open parentheses begin display style fraction numerator n x over denominator 2 end fraction end style close parentheses end fraction close square brackets open parentheses D i v i d i n g space n u m e r a t o r space a n d space d e n o m i n a t o r space b y space x squared close parentheses comma equals limit as x rightwards arrow 0 of open square brackets fraction numerator begin display style fraction numerator sin squared begin display style fraction numerator m x over denominator 2 end fraction end style over denominator x squared end fraction end style over denominator begin display style fraction numerator sin squared begin display style fraction numerator n x over denominator 2 end fraction end style over denominator x squared end fraction end style end fraction close square brackets equals limit as x rightwards arrow 0 of open square brackets fraction numerator begin display style fraction numerator sin begin display style fraction numerator m x over denominator 2 end fraction end style over denominator x end fraction end style cross times begin display style fraction numerator sin begin display style fraction numerator m x over denominator 2 end fraction end style over denominator x end fraction end style over denominator begin display style fraction numerator sin begin display style fraction numerator n x over denominator 2 end fraction end style over denominator x end fraction end style cross times begin display style fraction numerator sin begin display style fraction numerator n x over denominator 2 end fraction end style over denominator x end fraction end style end fraction close square brackets equals limit as x rightwards arrow 0 of open square brackets fraction numerator begin display style fraction numerator sin begin display style fraction numerator m x over denominator 2 end fraction end style over denominator begin display style fraction numerator m x over denominator 2 end fraction end style end fraction end style cross times begin display style fraction numerator sin begin display style fraction numerator m x over denominator 2 end fraction end style over denominator begin display style fraction numerator m x over denominator 2 end fraction end style end fraction end style cross times begin display style m squared over 4 end style over denominator begin display style fraction numerator sin begin display style fraction numerator n x over denominator 2 end fraction end style over denominator begin display style fraction numerator n x over denominator 2 end fraction end style end fraction end style cross times begin display style fraction numerator sin begin display style fraction numerator n x over denominator 2 end fraction end style over denominator begin display style fraction numerator n x over denominator 2 end fraction end style end fraction end style cross times begin display style n squared over 4 end style end fraction close square brackets equals fraction numerator 1 cross times 1 over denominator 1 cross times 1 end fraction cross times m squared over n squared equals m squared over n squared$

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