Which of the following limit is/are equal to unity?

$l i m x \to 0^{+} ((t a n x)^{s i n x} + (s i n x)^{t a n x})$

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$l i m x \to 0^{+} \frac{\sqrt{1 - e^{- x}} - \sqrt{1 - c o s x}}{\sqrt{s i n x}}$

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$l i m x \to 0 ([1 + | x |])^{\frac{1}{| x |}}$ (where [.] is G.I.F.)

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$l i m x \to \infty \frac{n + 1 \sum k = 1 k^{2}}{n \sum k = 1 k^{2}}; n ϵ N$

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Solution

The correct options are
B
$l i m x \to 0^{+} \frac{\sqrt{1 - e^{- x}} - \sqrt{1 - c o s x}}{\sqrt{s i n x}}$

C
$l i m x \to 0 ([1 + | x |])^{\frac{1}{| x |}}$ (where [.] is G.I.F.)

D
$l i m x \to \infty \frac{n + 1 \sum k = 1 k^{2}}{n \sum k = 1 k^{2}}; n ϵ N$

(A) $l i m x \to 0^{+} (e^{s i n x I n (t a n x)}) + e^{t a n x I n (s i n x)}$
$l i m x \to 0^{+} (e^{\frac{s i n x}{x}} . \frac{I n (t a n x)}{\frac{1}{x}} + e^{\frac{t a n x I n}{x}} \frac{I n (s i n x)}{\frac{1}{x}}) = 1 + 1 = 2$
(B) $l i m x \to 0^{+} \frac{1 - e^{- x} - 1 + c o s x}{\sqrt{s i n x} (\sqrt{1 - e^{- x}} + \sqrt{1 - c o s x})}$
$l i m x \to 0^{+} \frac{1 - \frac{x^{2}}{2!} + . . . ., - (1 - \frac{x}{1!} + \frac{x^{2}}{2!} . . . . .)}{\sqrt{x} \sqrt{\frac{s i n x}{x}} (\sqrt{1 - e^{- x}} + \sqrt{1 - c o s x})} = 1$

(C) Base is exact 1

(D) $l i m x \to \infty \frac{(n + 1) (n + 2) (2 n + 3)}{6. \frac{n (n + 1) (2 n + 1)}{6}} = 1$

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