The value of `lim_(x to 0) ((4^(x) - 1)^(3))/("sin"(x)/(4) log (1 + (x^(2))/(3)))` equals

Options

A. `3 (log 4)^(3)`
B. `4 (log 4)^(3)`
C. `12 (log4)^(3)`
D. `15 (log4)^(3)`

Answer

To solve the limit

\[ \lim_{x \to 0} \frac{(4^x - 1)^3}{\frac{\sin(x)}{4} \log\left(1 + \frac{x^2}{3}\right)}, \]

we can follow these steps:

### Step 1: Identify Basic Limits
Recall the following basic limits:
1. \(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
2. \(\lim_{x \to 0} \frac{a^x - 1}{x} = \log a\)
3. \(\lim_{x \to 0} \frac{\log(1 + x)}{x} = 1\)

### Step 2: Rewrite the Limit
We can rewrite the limit in a form that allows us to apply these basic limits.

We start with the expression:

\[ (4^x - 1)^3 \]

Using the second basic limit, we can express \(4^x - 1\) as:

\[ 4^x - 1 = 4^x - 1 = (4^x - 1) \cdot \frac{x}{x} = x \cdot \frac{4^x - 1}{x}. \]

Thus,

\[ (4^x - 1)^3 = \left(x \cdot \frac{4^x - 1}{x}\right)^3 = x^3 \left(\frac{4^x - 1}{x}\right)^3. \]

### Step 3: Rewrite the Denominator
Now, for the denominator:

\[ \frac{\sin(x)}{4} \log\left(1 + \frac{x^2}{3}\right) = \frac{1}{4} \sin(x) \cdot \log\left(1 + \frac{x^2}{3}\right). \]

Using the first basic limit, we can express \(\sin(x)\) as:

\[ \sin(x) = x \cdot \frac{\sin(x)}{x}. \]

And for \(\log\left(1 + \frac{x^2}{3}\right)\):

\[ \log\left(1 + \frac{x^2}{3}\right) = \frac{x^2}{3} \cdot \frac{\log\left(1 + \frac{x^2}{3}\right)}{\frac{x^2}{3}}. \]

### Step 4: Substitute Back into the Limit
Now substituting these back into the limit gives:

\[ \lim_{x \to 0} \frac{x^3 \left(\frac{4^x - 1}{x}\right)^3}{\frac{1}{4} \cdot x \cdot \frac{\sin(x)}{x} \cdot \frac{x^2}{3} \cdot \frac{\log\left(1 + \frac{x^2}{3}\right)}{\frac{x^2}{3}}}. \]

### Step 5: Simplify the Limit
This simplifies to:

\[ \lim_{x \to 0} \frac{4^x - 1}{x}^3 \cdot \frac{4}{\sin(x)} \cdot \frac{3}{x^2} \cdot \log\left(1 + \frac{x^2}{3}\right). \]

### Step 6: Evaluate the Limit
Now we can evaluate each part:

1. \(\lim_{x \to 0} \frac{4^x - 1}{x} = \log 4\).
2. \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\).
3. \(\lim_{x \to 0} \frac{\log\left(1 + \frac{x^2}{3}\right)}{\frac{x^2}{3}} = 1\).

Putting it all together:

\[ \lim_{x \to 0} \frac{(4^x - 1)^3}{\frac{\sin(x)}{4} \log\left(1 + \frac{x^2}{3}\right)} = \frac{(\log 4)^3}{\frac{1}{4} \cdot 1 \cdot 1} = 12 (\log 4)^3. \]

### Final Answer
Thus, the value of the limit is:

\[ 12 (\log 4)^3. \]