Consider the function: `f(x)={{:((tankx)/(x),xlt0),(3x+2k^(2),xle0):}`
What is the non-zero value of k for which the function is continuous at x = 0?

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. 1
D. 2

Answer

To determine the non-zero value of \( k \) for which the function

\[ f(x) = \begin{cases} \frac{\tan(kx)}{x} & \text{if } x < 0 \\ 3x + 2k^2 & \text{if } x \geq 0 \end{cases} \]

is continuous at \( x = 0 \), we need to ensure that the left-hand limit and the right-hand limit at \( x = 0 \) are equal.

### Step 1: Calculate the Left-Hand Limit as \( x \to 0^- \)

The left-hand limit is given by:

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{\tan(kx)}{x} \]

Substituting \( x = 0 \) directly gives us \( \frac{0}{0} \), which is an indeterminate form. We can apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule

Using L'Hôpital's Rule, we differentiate the numerator and denominator:

\[ \lim_{x \to 0^-} \frac{\tan(kx)}{x} = \lim_{x \to 0^-} \frac{k \sec^2(kx)}{1} \]

Now substituting \( x = 0 \):

\[ = k \sec^2(0) = k \cdot 1 = k \]

Thus, the left-hand limit is:

\[ \lim_{x \to 0^-} f(x) = k \]

### Step 3: Calculate the Right-Hand Limit as \( x \to 0^+ \)

The right-hand limit is given by:

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (3x + 2k^2) \]

Substituting \( x = 0 \):

\[ = 3(0) + 2k^2 = 2k^2 \]

Thus, the right-hand limit is:

\[ \lim_{x \to 0^+} f(x) = 2k^2 \]

### Step 4: Set the Left-Hand Limit Equal to the Right-Hand Limit

For the function to be continuous at \( x = 0 \), we set the left-hand limit equal to the right-hand limit:

\[ k = 2k^2 \]

### Step 5: Rearranging the Equation

Rearranging gives us:

\[ 2k^2 - k = 0 \]

Factoring out \( k \):

\[ k(2k - 1) = 0 \]

### Step 6: Solve for \( k \)

Setting each factor to zero gives:

1. \( k = 0 \)
2. \( 2k - 1 = 0 \) which leads to \( k = \frac{1}{2} \)

Since we are looking for the non-zero value of \( k \), we have:

\[ k = \frac{1}{2} \]

### Conclusion

The non-zero value of \( k \) for which the function is continuous at \( x = 0 \) is:

\[ \boxed{\frac{1}{2}} \]