If $f(x)=\frac{x\left(a^x-1\right)}{1-\cos x}$ and $g(x)=\frac{x\left(1-a^x\right)}{a^x\left(\sqrt{1-x^2}-\sqrt{1+x^2}\right)}$, then $\lim _{x \rightarrow 0}(f(x)-g(x))=$

• A

  $3 \log a$
• B

  $e^a$
• C

  $2 \log a$
• Correct answer
  D

  $\log a$

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