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Answer 1

EXPLANATION.

$\sf \implies \displaystyle \lim_{x \to 0} \bigg(\dfrac{x^{2} }{sinx^{2} } \bigg)$

As we know that,

Formula of :

$\sf \implies \displaystyle \lim_{x \to 0} \bigg(\dfrac{x}{sin(x)} \bigg) = 1.$

Using this formula in the equation.

We can write equation as,

$\sf \implies \displaystyle \lim_{x \to 0} \bigg( \dfrac{x}{sin(x)} \bigg)^{2} = (1)^{2} = 1.$

$\sf \implies \displaystyle \lim_{x \to 0} \bigg(\dfrac{x^{2} }{sinx^{2} } \bigg) = 1.$

                                                                                                                 

MORE INFORMATION.

(1) = eˣ = 1 + x + x²/2! + x³/3! + . . . . .

(2) = e⁻ˣ = 1 - x + x²/2! - x³/3! + . . . . .

(3) = ㏒(1 + x) = x - x²/2 + x³/3 - . . . . .

(4) = ㏒(1 - x) = - x - x²/2 - x³/3 - . . . . .

(5) = aˣ = 1 + (x ㏒ a) + (x ㏒ a)²/2! + (x ㏒ a)³/3! + . . . . .

(6) = sin x = x - x³/3! + x⁵/5! - . . . . .

(7) = cos x = 1 - x²/2! + x⁴/4! - . . . . .

(8) = tan x = x + x³/3 + 2x⁵/15 + . . . . .

Answer 2

${\orange{ \boxed{ \boxed{ \begin{array}{cc} \bf \: \hookrightarrow \: given \: : \\ \\ \rm \displaystyle\lim_{ x \to 0} \rm \: \frac{ \green{{x}^{2}} }{sin \: \green{{x}^{2} }} \\ \\ {\pink{ { \boxed{ \begin{array}{cc} \sf \: we \: know \: that : \\ \\\sf \hookrightarrow \: \rm\displaystyle\lim_{ x \to 0} \rm \: \frac{ \blue{x}}{sin \: \blue{ x}} = 1 \\ \\ \rm \hookrightarrow \: \rm \displaystyle\lim_{ x \to 0} \rm \frac{sin \: x}{x} = 1\end{array}}}}} \\ \\ \sf \: according \: to \: the \: question \: \\ \\ \rm \: \green{{x}^{2} } \: denotes \: = \blue{ x} \\ \\ \bf \: so \\ \\ = \rm \displaystyle\lim_{ x \to 0} \rm \frac{ \green{ {x}^{2} }}{sin \: \green{ {x}^{2}} } \\ \\ = 1 \\ \\ \boxed{\sf \: final \: answer \: :\rm \displaystyle\lim_{ x \to 0} \rm \frac{ {x}^{2} }{sin \: {x}^{2} } = 1} \end{array}}}}}$

Remember :

$\sf \hookrightarrow \: \rm \: sin \: {x}^{2} \neq \: {(sin \: x)}^{2} \\ \\ \sf \hookrightarrow \: \rm \: ( {sin \: x)}^{2} = {sin}^{2} \: x$

So,

$\rm \frac{ {x}^{2} }{sin \: {x}^{2} } \neq \: ({ \frac{x}{sin \: x} })^{2}$