Discuss the continuity of the following functions :
`f(x)=sinx/cosx`.

Answer

To discuss the continuity of the function \( f(x) = \frac{\sin x}{\cos x} \), we can follow these steps:

### Step 1: Identify the function
The given function can be rewritten as:
\[ f(x) = \tan x \]
This is because \( \tan x = \frac{\sin x}{\cos x} \).

### Step 2: Determine where the function is defined
The function \( f(x) = \tan x \) is defined wherever the denominator \( \cos x \) is not equal to zero. Therefore, we need to find the values of \( x \) for which \( \cos x = 0 \).

### Step 3: Find the points where \( \cos x = 0 \)
The cosine function is equal to zero at:
\[ x = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \]
This means that \( \cos x = 0 \) at odd multiples of \( \frac{\pi}{2} \), specifically at:
\[ x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots \]

### Step 4: Define the intervals of continuity
Since \( f(x) \) is undefined at these points, we can say that the function is continuous everywhere else. Therefore, the function \( f(x) \) is continuous on the intervals:
\[ (-\infty, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \frac{3\pi}{2}) \cup (\frac{3\pi}{2}, \frac{5\pi}{2}) \cup \ldots \]
In a more general form, we can express this as:
\[ f(x) \text{ is continuous for } x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + n\pi \, | \, n \in \mathbb{Z} \right\} \]

### Conclusion
Thus, the function \( f(x) = \frac{\sin x}{\cos x} \) is continuous for all \( x \) in \( \mathbb{R} \) except at the points where \( \cos x = 0 \).

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