Question

The value of $\underset{x\to 0}{\lim }\frac{15^x-5^x-3^x+1}{1-cos2x}$ is

• A. $\frac{\left(log3\right)\left(log5\right)}{2}$
• B. $2(log3)(log5)$
• C. $\frac{log3+log5}{2}$
• D. None of these

Answer 1

Answer: (A)

$\underset{x\to 0}{\lim }\frac{15^x-5^x-3^x+1}{1-\cos 2x}$
$=\underset{x\to 0}{\lim }\frac{\left(3^x-1\right)\left(5^x-1\right)}{1-1+2{\sin }^{2}x}$
$=\underset{x\to 0}{\lim }\left(\frac{3^x-1}{x}\right)\left(\frac{5^x-1}{x}\right)\left(\frac{x^2}{2{\sin }^{2}x}\right)$
$=\frac{1}{2}(\log 3)(\log 5)\cdot 1$
$\left[\because \underset{x\to 0}{\lim }\left(\frac{a^x-b^x}{x}\right)=\log \left(\frac{a}{b}\right)\right]$