Q53 / 150 · MHT CET 2010 · Mathematics

The value of $\lim _{x \rightarrow 0} \frac{15^{x}-5^{x}-3^{x}+1}{1-\cos 2 x}$ is

  • Correct answer
    A
    $\frac{(\log 3)(\log 5)}{2}$
  • B
    $2(\log 3)(\log 5)$
  • C
    $\frac{\log 3+\log 5}{2}$
  • D
    None of these

Solution

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

Hence, the correct answer is option: 'A'.

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