If : f(x) `{:(=(sinx )/x+ cos x", ..."x!=0),(=2", ... "x=0):}` then :

Options

A. `lim_(x to 0^(+)) f(x) =2 `
B. ` lim _(x to 0^(-)) f(x) = 0 `
C. f(x) is continuus at x = 0
D. none of these

Answer

To determine if the function \( f(x) \) is continuous at \( x = 0 \), we need to check the following conditions:

1. **Find \( f(0) \)**.
2. **Calculate \( \lim_{x \to 0} f(x) \)**.
3. **Check if \( \lim_{x \to 0} f(x) = f(0) \)**.

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

From the definition of the function:
\[ f(x) = \begin{cases} \frac{\sin x}{x} + \cos x & \text{if } x \neq 0 \\ 2 & \text{if } x = 0 \end{cases} \]
Thus,
\[ f(0) = 2. \]

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

Since we are interested in the limit as \( x \) approaches 0, we will consider the case when \( x \neq 0 \):
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \left( \frac{\sin x}{x} + \cos x \right). \]

We know from calculus that:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1, \]
and
\[ \cos(0) = 1. \]

Therefore,
\[ \lim_{x \to 0} f(x) = 1 + 1 = 2. \]

### Step 3: Check if \( \lim_{x \to 0} f(x) = f(0) \)

Now we compare the limit we calculated with the value of the function at 0:
\[ \lim_{x \to 0} f(x) = 2 \quad \text{and} \quad f(0) = 2. \]

Since both values are equal, we conclude that:
\[ \lim_{x \to 0} f(x) = f(0). \]

### Conclusion

Since all conditions for continuity at \( x = 0 \) are satisfied, we can conclude that \( f(x) \) is continuous at \( x = 0 \).

### Final Answer
The function \( f(x) \) is continuous at \( x = 0 \).

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