` f(x)={{:(3x-8, if x le 5),(2k, if x gt 5) :}` at `x = 5`

Explanation

To solve the problem, we need to find the value of \( k \) such that the function \( f(x) \) is continuous at \( x = 5 \). The function is defined as:

\[ f(x) = \begin{cases} 3x - 8 & \text{if } x \leq 5 \\ 2k & \text{if } x > 5 \end{cases} \]

### Step 1: Find the left-hand limit as \( x \) approaches 5

The left-hand limit is given by:

\[ \lim_{x \to 5^-} f(x) = f(5) = 3(5) - 8 \]

Calculating this gives:

\[ f(5) = 15 - 8 = 7 \]

### Step 2: Find the right-hand limit as \( x \) approaches 5

The right-hand limit is given by:

\[ \lim_{x \to 5^+} f(x) = 2k \]

### Step 3: Set the left-hand limit equal to the right-hand limit

For the function to be continuous at \( x = 5 \), the left-hand limit must equal the right-hand limit:

\[ 7 = 2k \]

### Step 4: Solve for \( k \)

To find \( k \), we rearrange the equation:

\[ k = \frac{7}{2} \]

### Final Answer

Thus, the value of \( k \) is:

\[ k = \frac{7}{2} \]

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