If the function \( f(x)=\displaystyle\frac{(4^x-1)^3}{\sin\frac{x}{4}\cdot\log\left(1+\frac{x^2}{3}\right)} \) is continuous at \( x=0 \), then the value of \( f(0) \) is: (Take \( \log_e x=\log x \))
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 12(\log 4)^3 \)
For continuity at x = 0, \( f(0)=\lim_{x\to 0}f(x) \). The limit follows from the standard limits
\( \displaystyle \lim_{x\to 0}\frac{a^x-1}{x}=\log a \)
\( \displaystyle \lim_{x\to 0}\frac{\sin x}{x}=1 \)
\( \displaystyle \lim_{x\to 0}\frac{\log(1+x)}{x}=1 \)
Since \( \frac{x}{4}\cdot\frac{x^2}{3}=\frac{x^3}{12} \), each factor can be put in one of these forms:
\( \displaystyle f(x)=\frac{12\left(\frac{4^x-1}{x}\right)^3}{\frac{\sin(x/4)}{x/4}\cdot\frac{\log(1+x^2/3)}{x^2/3}} \)
Taking the limit,
\( \displaystyle \lim_{x\to 0}f(x)=\frac{12(\log 4)^3}{1\cdot 1} \)
\( \displaystyle =12(\log 4)^3 \)
So \( f(0)=12(\log 4)^3 \).
Why the other options are incorrect
Option (1) \( \displaystyle\frac{1}{12}(\log 4)^3 \) — Incorrect. \( \frac{1}{12}(\log 4)^3 \) comes from multiplying by x/4 and x2/3 instead of dividing by them.
Option (3) \( \log 4 \) — Incorrect. The limit is \( 12(\log 4)^3 \), not log 4.
Option (4) 12 — Incorrect. The factor \( (\log 4)^3 \) from \( \lim\frac{4^x-1}{x}=\log 4 \) is missing.