_ x 0 4^x - 9^x x( 4^x + 9^x ) = - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to 0} \frac{{{4^x} - {9^x}}}{{x({4^x} + {9^x})}} = $

✓
$\log \left( {\frac{2}{3}} \right)$
B
$\frac{1}{2}\log \left( {\frac{3}{2}} \right)$
C
$\frac{1}{2}\log \left( {\frac{2}{3}} \right)$
D
$\log \,\left( {\frac{3}{2}} \right)$

✓

Answer

Correct option: A.

$\log \left( {\frac{2}{3}} \right)$

a
(a) $y = \mathop {\lim }\limits_{x \to 0} \frac{{{4^x} - {9^x}}}{{x({4^x} + {9^x})}}$,$\left( {\frac{0}{0}{\rm{form}}} \right)$

Using $L-$ Hospital’s rule,

$y = \mathop {\lim }\limits_{x \to 0} \frac{{{4^x}\log 4 - {9^x}\log 9}}{{({4^x} + {9^x}) + x({4^x}\log 4 + {9^x}\log 9)}}$

==> $y = \frac{{\log 4 - \log 9}}{2}$

==> $y = \frac{{\log {{\left( {\frac{2}{3}} \right)}^2}}}{2} = \log \frac{2}{3}$.

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