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

MCQ

$\lim _{x \rightarrow 0} \frac{4^x-9^x}{x\left(4^x+9^x\right)}=$

✓
$\log \left(\frac{2}{3}\right)$
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)
Applying L-Hospital's rule,
$\lim _{x \rightarrow 0} \frac{4^x-9^x}{x\left(4^x+9^x\right)}=\lim _{x \rightarrow 0} \frac{4^x \log 4-9^x \log 9}{\left(4^x+9^x\right)+x\left(4^x \log 4+9^x \log 9\right)}$
$=\frac{\log 4-\log 9}{2}=\frac{\log \left(\frac{2}{3}\right)^2}{2}$
$=\log \frac{2}{3}$

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