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Which of the following is correct. (Where[.] is G.I.F)

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$\lim_{x \to 0} \frac{[e^{x}]}{[e^{x}]} = 1$

$\lim_{x \to \frac{π}{2}} \cos^{- 1} [\cot x] = \frac{π}{2}$

$\lim_{x \to 0} \frac{x}{x} = 1$

$\lim_{x \to 0} \frac{[x]}{[x]} = 1$

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Detailed Solution

A) $x \to 0^{+} \Rightarrow x > 0 \Rightarrow e^{x} > 1 [e^{x}] = 1$ $R . H . L = 1 x < 0^{-} \Rightarrow x < 0 e^{x} < 1 [e^{x}] = 0$ L.H.L= indeterminate

B) $x \to \frac{π^{+}}{2} \Rightarrow \cot x < \cot \frac{π}{2} \Rightarrow [\cot x] = - 1$ $∴ R . H . L = π L . H . L = \frac{π}{2}$

C) $\lim_{x \to 0} \frac{x}{x} = 1$

D) Limit does not exist

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