Find the value of f(0) so that the function `f(x)=1/24 (4^x-1)^3/(sin (x/4) log (1+x^2/3)(log 2)^3), x ne 0` is continuous in `R` is .

Answer

To find the value of \( f(0) \) so that the function

\[ f(x) = \frac{1}{24} \cdot \frac{(4^x - 1)^3}{\sin\left(\frac{x}{4}\right) \cdot \log\left(1 + \frac{x^2}{3}\right) \cdot (\log 2)^3} \]

is continuous in \( \mathbb{R} \), we need to ensure that the limit of \( f(x) \) as \( x \) approaches 0 is equal to \( f(0) \).

### Step 1: Find the limit as \( x \to 0 \)

We start by calculating

\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \left( \frac{1}{24} \cdot \frac{(4^x - 1)^3}{\sin\left(\frac{x}{4}\right) \cdot \log\left(1 + \frac{x^2}{3}\right) \cdot (\log 2)^3} \right). \]

### Step 2: Simplify the limit

Since \( \frac{1}{24} \) and \( (\log 2)^3 \) are constants, we can factor them out of the limit:

\[ \lim_{x \to 0} f(x) = \frac{1}{24 (\log 2)^3} \lim_{x \to 0} \frac{(4^x - 1)^3}{\sin\left(\frac{x}{4}\right) \cdot \log\left(1 + \frac{x^2}{3}\right)}. \]

### Step 3: Evaluate \( (4^x - 1)^3 \)

Using the property \( \lim_{x \to 0} \frac{a^x - 1}{x} = \log a \):

\[ \lim_{x \to 0} \frac{4^x - 1}{x} = \log 4. \]

Thus,

\[ \lim_{x \to 0} (4^x - 1) = x \cdot \log 4. \]

So,

\[ \lim_{x \to 0} (4^x - 1)^3 = \lim_{x \to 0} (x \cdot \log 4)^3 = (\log 4)^3 \cdot x^3. \]

### Step 4: Evaluate \( \sin\left(\frac{x}{4}\right) \)

Using the property \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \):

\[ \lim_{x \to 0} \sin\left(\frac{x}{4}\right) = \frac{x}{4}. \]

### Step 5: Evaluate \( \log\left(1 + \frac{x^2}{3}\right) \)

Using the property \( \lim_{x \to 0} \frac{\log(1 + x)}{x} = 1 \):

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

### Step 6: Substitute back into the limit

Now substituting these limits back into our expression:

\[ \lim_{x \to 0} f(x) = \frac{1}{24 (\log 2)^3} \cdot \frac{(\log 4)^3 \cdot x^3}{\left(\frac{x}{4}\right) \cdot \left(\frac{x^2}{3}\right)}. \]

### Step 7: Simplify the expression

This simplifies to:

\[ = \frac{1}{24 (\log 2)^3} \cdot \frac{(\log 4)^3 \cdot x^3}{\frac{x^3}{12}} = \frac{1}{24 (\log 2)^3} \cdot (\log 4)^3 \cdot 12. \]

### Step 8: Final simplification

Thus,

\[ \lim_{x \to 0} f(x) = \frac{12 \cdot (\log 4)^3}{24 \cdot (\log 2)^3} = \frac{(\log 4)^3}{2 \cdot (\log 2)^3}. \]

Since \( \log 4 = 2 \log 2 \):

\[ = \frac{(2 \log 2)^3}{2 \cdot (\log 2)^3} = \frac{8 (\log 2)^3}{2 (\log 2)^3} = 4. \]

### Conclusion

Therefore, to make \( f(x) \) continuous at \( x = 0 \), we set:

\[ f(0) = 4. \]