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Maths-

General

Easy

Question

$L t subscript x not stretchy rightwards arrow 0 end subscript fraction numerator x tan invisible function application 2 x minus 2 x tan invisible function application x over denominator left parenthesis 1 minus cos invisible function application 2 x right parenthesis squared end fraction$

1
$\frac{1}{2}$
0
2

The correct answer is: $1 half$

$limit as x rightwards arrow 0 of open square brackets fraction numerator x tan 2 x minus 2 x tan x over denominator open parentheses 1 minus cos 2 x close parentheses squared end fraction close square brackets equals limit as x rightwards arrow 0 of open square brackets fraction numerator x open parentheses begin display style fraction numerator 2 tan x over denominator 1 minus tan squared x end fraction end style close parentheses minus 2 x tan x over denominator open parentheses 2 sin squared x close parentheses squared end fraction close square brackets equals limit as x rightwards arrow 0 of open square brackets fraction numerator begin display style fraction numerator 2 x tan x minus 2 x tan x plus 2 x tan cubed x over denominator open parentheses 1 minus tan squared x close parentheses end fraction end style over denominator 4 sin to the power of 4 x end fraction close square brackets equals limit as x rightwards arrow 0 of open square brackets fraction numerator 2 x tan cubed x over denominator open parentheses 1 minus tan squared x close parentheses open parentheses 4 sin to the power of 4 x close parentheses end fraction close square brackets equals limit as x rightwards arrow 0 of open square brackets fraction numerator 1 over denominator open parentheses 1 minus tan squared x close parentheses end fraction cross times fraction numerator x tan cubed x over denominator 2 sin to the power of 4 x end fraction close square brackets equals limit as x rightwards arrow 0 of open square brackets fraction numerator 1 over denominator open parentheses 1 minus tan squared x close parentheses end fraction cross times fraction numerator x sin cubed x over denominator 2 sin to the power of 4 x cos cubed x end fraction close square brackets equals limit as x rightwards arrow 0 of open square brackets fraction numerator 1 over denominator open parentheses 1 minus tan squared x close parentheses end fraction cross times fraction numerator x over denominator 2 sin x cos cubed x end fraction close square brackets equals limit as x rightwards arrow 0 of open square brackets fraction numerator 1 over denominator open parentheses 1 minus tan squared x close parentheses end fraction cross times fraction numerator 1 over denominator begin display style fraction numerator sin x over denominator x end fraction end style end fraction cross times fraction numerator 1 over denominator 2 cos cubed x end fraction close square brackets equals fraction numerator 1 over denominator open parentheses 1 minus tan squared open parentheses 0 close parentheses close parentheses end fraction cross times 1 over 1 cross times 1 half equals 1 half$

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