The value of _ Limit _ x , , ^ - 1 ( x^ - a , , _a x ) ^ - 1 ( a^… - Vidyadip

MCQ

The value of $\mathop {_{Limit}}\limits_{x \to \,\infty } \,\frac{{{{\cot }^{ - 1}}\left( {{x^{ - a}}\,\,{{\log }_a}x} \right)}}{{{{\sec }^{ - 1}}\left( {{a^x}\,\,{{\log }_x}a} \right)}}$ $(a > 1)$ is equal to

✓
$1$
B
$0$
C
$\pi /2$
D
does not exist

✓

Answer

Correct option: A.

$1$

a
$\mathop {Limit}\limits_{x \to \infty } \,\frac{{{{\cot }^{ - 1}}\left( {\frac{{{{\log }_a}x}}{{{x^a}}}} \right)}}{{{{\sec }^{ - 1}}\left( {\frac{{{a^x}}}{{{{\log }_a}x}}} \right)}}\,$ ;

$\mathop {as}\limits_{x \to \infty } \,\left( {\frac{{{{\log }_a}x}}{{{x^a}}}} \right)\,\, \to \,\,0$ and $\,\left( {\frac{{{a^x}}}{{{{\log }_a}x}}} \right)\, \to \,\,\infty \,$ (using L’opital rule)

$\therefore l =$$\frac{{\pi /2}}{{\pi /2}}$ $= 1 $

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