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Maths-

General

Easy

Question

$Lt subscript x not stretchy rightwards arrow 0 end subscript fraction numerator cos a x minus cos b x over denominator x squared end fraction$

$\frac{a^{2} - b^{2}}{2}$
$\frac{b^{2} - a^{2}}{2}$
$\frac{a^{2} + b^{2}}{2}$
$2 (a^{2} - b^{2})$

The correct answer is: $fraction numerator b squared minus a squared over denominator 2 end fraction$

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

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