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Maths-

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Question

The set of values of x for which the inequality $blank s i n to the power of 4 end exponent invisible function application open parentheses fraction numerator x over denominator 3 end fraction close parentheses plus c o s to the power of 4 end exponent invisible function application open parentheses fraction numerator x over denominator 3 end fraction close parentheses less or equal than fraction numerator 1 over denominator 2 end fraction text end text$ holds is

$R$
$\{x / x = \frac{3 n π}{2} \pm \frac{3 π}{4}; n \in l\}$
$R - \{x / x = \frac{3 n π}{2} \pm \frac{3 π}{4}; n \in l\}$
$ϕ$

The correct answer is: $R minus open curly brackets x divided by x equals fraction numerator 3 n pi over denominator 2 end fraction plus-or-minus fraction numerator 3 pi over denominator 4 end fraction semicolon n element of l close curly brackets$

Given inequality is
$s i n to the power of 4 end exponent invisible function application open parentheses fraction numerator x over denominator 3 end fraction close parentheses plus c o s to the power of 4 end exponent invisible function application open parentheses fraction numerator x over denominator 3 end fraction close parentheses less or equal than fraction numerator 1 over denominator 2 end fraction$
$rightwards double arrow 1 minus fraction numerator 1 over denominator 2 end fraction s i n to the power of 2 end exponent invisible function application fraction numerator 2 x over denominator 3 end fraction less or equal than fraction numerator 1 over denominator 2 end fraction$
$rightwards double arrow s i n to the power of 2 end exponent invisible function application fraction numerator 2 x over denominator 3 end fraction greater or equal than 1$
$text But end text s i n to the power of 2 end exponent invisible function application fraction numerator 2 x over denominator 3 end fraction less or equal than 1 text and end text s i n to the power of 2 end exponent invisible function application fraction numerator 2 x over denominator 3 end fraction equals 1 rightwards double arrow fraction numerator 2 x over denominator 3 end fraction equals n pi plus-or-minus fraction numerator pi over denominator 2 end fraction comma n element of I$
$text Therefore, end text$
$x element of R minus open curly brackets x ∣ x equals fraction numerator 3 n pi over denominator 2 end fraction plus-or-minus fraction numerator 3 pi over denominator 4 end fraction comma n element of 1 close curly brackets$

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$I f blank stack lim with x rightwards arrow infinity below invisible function application open parentheses fraction numerator x plus h over denominator x minus h end fraction close parentheses to the power of x end exponent equals 4 comma blank t h e n blank h equals$

$I f blank stack lim with x rightwards arrow infinity below invisible function application open parentheses fraction numerator x plus h over denominator x minus h end fraction close parentheses to the power of x end exponent equals 4 comma blank t h e n blank h equals$

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Arrange the following limits in descending order
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ii) $stack lim with x rightwards arrow 0 below invisible function application fraction numerator left parenthesis 1 plus x right parenthesis to the power of 1 divided by 8 end exponent minus left parenthesis 1 minus x right parenthesis to the power of 1 divided by 8 end exponent over denominator x end fraction$
iii) $stack lim with x rightwards arrow 0 below invisible function application fraction numerator t a n x minus s i n x over denominator x to the power of 3 end exponent end fraction$
iv) $stack lim with x rightwards arrow infinity below invisible function application open curly brackets square root of x to the power of 2 end exponent plus 6 x plus 7 end root minus x close curly brackets$

Arrange the following limits in descending order
i) $stack lim with x rightwards arrow 3 to the power of minus end exponent below invisible function application fraction numerator 3 minus x over denominator open vertical bar x minus 3 close vertical bar end fraction$
ii) $stack lim with x rightwards arrow 0 below invisible function application fraction numerator left parenthesis 1 plus x right parenthesis to the power of 1 divided by 8 end exponent minus left parenthesis 1 minus x right parenthesis to the power of 1 divided by 8 end exponent over denominator x end fraction$
iii) $stack lim with x rightwards arrow 0 below invisible function application fraction numerator t a n x minus s i n x over denominator x to the power of 3 end exponent end fraction$
iv) $stack lim with x rightwards arrow infinity below invisible function application open curly brackets square root of x to the power of 2 end exponent plus 6 x plus 7 end root minus x close curly brackets$

$I f blank open square brackets x close square brackets d e n o t e s blank t h e blank g r e a t e s t blank i n t e g e r blank l e s s blank t h a n blank o r blank e q u a l blank t o blank x blank t h e n blank$ $stack lim with n rightwards arrow infinity below invisible function application fraction numerator 1 over denominator n to the power of 4 end exponent end fraction open curly brackets open square brackets 1 to the power of 3 end exponent x close square brackets plus open square brackets 2 to the power of 3 end exponent x close square brackets plus open square brackets 3 to the power of 3 end exponent x close square brackets plus horizontal ellipsis plus left square bracket n to the power of 3 end exponent x right square bracket close curly brackets equals$

$I f blank open square brackets x close square brackets d e n o t e s blank t h e blank g r e a t e s t blank i n t e g e r blank l e s s blank t h a n blank o r blank e q u a l blank t o blank x blank t h e n blank$ $stack lim with n rightwards arrow infinity below invisible function application fraction numerator 1 over denominator n to the power of 4 end exponent end fraction open curly brackets open square brackets 1 to the power of 3 end exponent x close square brackets plus open square brackets 2 to the power of 3 end exponent x close square brackets plus open square brackets 3 to the power of 3 end exponent x close square brackets plus horizontal ellipsis plus left square bracket n to the power of 3 end exponent x right square bracket close curly brackets equals$

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parallel

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$stack lim with x rightwards arrow 0 below invisible function application fraction numerator open parentheses a b close parentheses to the power of x end exponent minus a to the power of x end exponent minus b to the power of x end exponent plus 1 over denominator x to the power of 2 end exponent end fraction equals$

$stack lim with x rightwards arrow 0 below invisible function application fraction numerator open parentheses a b close parentheses to the power of x end exponent minus a to the power of x end exponent minus b to the power of x end exponent plus 1 over denominator x to the power of 2 end exponent end fraction equals$

$stack lim with x rightwards arrow infinity below invisible function application fraction numerator x over denominator square root of 4 x to the power of 2 end exponent plus 1 minus 1 end root end fraction equals$

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