Is the function f (x) `= ( 3 x + 4 tan x)/( x) ` continuous at x = 0 ? If not, how many the function be defined to make it continuous at this point .

Answer

To determine if the function \( f(x) = \frac{3x + 4 \tan x}{x} \) is continuous at \( x = 0 \), we need to check the following:

1. **Check if \( f(0) \) is defined**:
\[ f(0) = \frac{3(0) + 4 \tan(0)}{0} \]
Since \( \tan(0) = 0 \), we have:
\[ f(0) = \frac{0 + 0}{0} = \frac{0}{0} \]
This is undefined. Therefore, \( f(x) \) is not defined at \( x = 0 \).

2. **Calculate the limit as \( x \) approaches 0**:
We need to find:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{3x + 4 \tan x}{x} \]
We can split this limit:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \left( \frac{3x}{x} + \frac{4 \tan x}{x} \right) = \lim_{x \to 0} (3 + 4 \cdot \frac{\tan x}{x}) \]

3. **Use the known limit**:
We know that:
\[ \lim_{x \to 0} \frac{\tan x}{x} = 1 \]
Therefore, we can substitute this into our limit:
\[ \lim_{x \to 0} f(x) = 3 + 4 \cdot 1 = 3 + 4 = 7 \]

4. **Conclusion about continuity**:
For \( f(x) \) to be continuous at \( x = 0 \), we need:
\[ f(0) = \lim_{x \to 0} f(x) \]
Since \( f(0) \) is undefined and the limit as \( x \) approaches 0 is 7, we can define \( f(0) \) to be 7 to make the function continuous at this point.

Thus, we can define the function as:
\[ f(x) = \begin{cases} \frac{3x + 4 \tan x}{x} & \text{if } x \neq 0 \\ 7 & \text{if } x = 0 \end{cases} \]

### Summary:
- The function is not continuous at \( x = 0 \) because it is not defined there.
- To make it continuous, we can define \( f(0) = 7 \).