Let `f(x)={{:(cosx, if ,x ge 0), (x+k, if ,x lt 0.):}`
Find the value of the constant k, given that `lim_(x to 0)f(x)` exists.

Answer

To solve the problem, we need to find the value of the constant \( k \) such that the limit \( \lim_{x \to 0} f(x) \) exists. The function \( f(x) \) is defined as follows:

\[ f(x) = \begin{cases} \cos x & \text{if } x \geq 0 \\ x + k & \text{if } x < 0 \end{cases} \]

### Step 1: Determine the limits from both sides as \( x \) approaches 0.

We need to calculate the left-hand limit \( \lim_{x \to 0^-} f(x) \) and the right-hand limit \( \lim_{x \to 0^+} f(x) \).

**Right-hand limit:**
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \cos x = \cos(0) = 1 \]

**Left-hand limit:**
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x + k) = 0 + k = k \]

### Step 2: Set the left-hand limit equal to the right-hand limit.

For the limit \( \lim_{x \to 0} f(x) \) to exist, the left-hand limit must equal the right-hand limit:
\[ k = 1 \]

### Conclusion

Thus, the value of the constant \( k \) is:
\[ \boxed{1} \]

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