If `f(x)={{:(((5^(x)-1)^(3))/(sin((x)/(a))log(1+(x^(2))/(3)))",",x ne0),(9(log_(e)5)^(3)",",x=0):}` is continuous at `x=0`, then value of 'a' equals

Options

A. 3
B. 2
C. 1
D. None of these

Answer

To determine the value of 'a' for which the function

\[ f(x) = \begin{cases} \frac{(5^x - 1)^3}{\sin\left(\frac{x}{a}\right) \log\left(1 + \frac{x^2}{3}\right)} & \text{if } x \neq 0 \\ 9(\log_e 5)^3 & \text{if } x = 0 \end{cases} \]

is continuous at \( x = 0 \), we need to ensure that

\[ \lim_{x \to 0} f(x) = f(0). \]

### Step 1: Calculate \( f(0) \)

From the definition of the function, we have:

\[ f(0) = 9(\log_e 5)^3. \]

### Step 2: Calculate \( \lim_{x \to 0} f(x) \)

We need to find:

\[ \lim_{x \to 0} \frac{(5^x - 1)^3}{\sin\left(\frac{x}{a}\right) \log\left(1 + \frac{x^2}{3}\right)}. \]

### Step 3: Analyze the numerator

Using the limit \( \lim_{x \to 0} \frac{5^x - 1}{x} = \log_e 5 \), we can express:

\[ 5^x - 1 \sim x \log_e 5 \text{ as } x \to 0. \]

Thus,

\[ (5^x - 1)^3 \sim (x \log_e 5)^3 = x^3 (\log_e 5)^3. \]

### Step 4: Analyze the denominator

For small \( x \), we have:

\[ \sin\left(\frac{x}{a}\right) \sim \frac{x}{a} \text{ and } \log\left(1 + \frac{x^2}{3}\right) \sim \frac{x^2}{3}. \]

Thus, the denominator becomes:

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

### Step 5: Substitute into the limit

Now substituting these approximations into the limit gives:

\[ \lim_{x \to 0} \frac{(5^x - 1)^3}{\sin\left(\frac{x}{a}\right) \log\left(1 + \frac{x^2}{3}\right)} \sim \lim_{x \to 0} \frac{x^3 (\log_e 5)^3}{\frac{x^3}{3a}} = \lim_{x \to 0} \frac{3a (\log_e 5)^3}{1} = 3a (\log_e 5)^3. \]

### Step 6: Set the limit equal to \( f(0) \)

For continuity at \( x = 0 \):

\[ 3a (\log_e 5)^3 = 9(\log_e 5)^3. \]

### Step 7: Solve for 'a'

Dividing both sides by \( (\log_e 5)^3 \) (which is non-zero), we get:

\[ 3a = 9 \implies a = 3. \]

### Conclusion

Thus, the value of \( a \) is

\[ \boxed{3}. \]