Question

Evaluate \[ \lim_{x\to 0}\frac{\log(4+x)^x-\log 4^x}{\sin^2 x}. \]

• A. \(4\)
• B. \(\dfrac14\)
• C. \(2\)
• D. \(\dfrac12\)

Hint:

For limits involving logarithms, first convert powers using \(\log(a^x)=x\log a\). Then try to create the standard limit \[ \lim_{t\to0}\frac{\log(1+t)}{t}=1. \] Whenever \(\sin x\) appears in the denominator, use the approximation \(\sin x \sim x\) near \(x=0\).

Answer 1

The Correct Option is B