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To solve this problem, we need to determine the slope of the tangent line to the parametric curve \( C \) at \( t = 0 \). The parametric equations for the curve are:

\[
x(t) = \frac{3t - 1}{t^2 + 1}, \quad y(t) = \frac{2t + 4}{1 + t^2}.
\]

### Step-by-Step Solution:

1. **Find the derivatives** \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) at \( t = 0 \), because the slope of the tangent line is given by \( \frac{dy/dt}{dx/dt} \).

#### Derivative of \( x(t) \):
\[
x(t) = \frac{3t - 1}{t^2 + 1}.
\]
Using the quotient rule for derivatives:
\[
\frac{dx}{dt} = \frac{(t^2 + 1)(3) - (3t - 1)(2t)}{(t^2 + 1)^2}.
\]
At \( t = 0 \):
\[
\frac{dx}{dt} = \frac{(1)(3) - (-1)(0)}{(1^2)} = 3.
\]

#### Derivative of \( y(t) \):
\[
y(t) = \frac{2t + 4}{1 + t^2}.
\]
Again, using the quotient rule:
\[
\frac{dy}{dt} = \frac{(1 + t^2)(2) - (2t + 4)(2t)}{(1 + t^2)^2}.
\]
At \( t = 0 \):
\[
\frac{dy}{dt} = \frac{(1)(2) - (4)(0)}{(1^2)} = 2.
\]

2. **Calculate the slope** of the tangent line at \( t = 0 \):
\[
\text{Slope} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2}{3}.
\]

Thus, the slope of the tangent line at \( t = 0 \) is \( \frac{2}{3} \).

### Final Answer:
The correct option is **\( \frac{2}{3} \)**.

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Would you like further details or have any questions? Here are some related questions you might find helpful:

1. How do you apply the quotient rule in calculus?
2. What is the geometric interpretation of the slope of a tangent line?
3. How can you find the second derivative of a parametric curve?
4. What is the meaning of parametric equations in calculus?
5. How would you generalize this process for finding the slope at any \( t \)?

**Tip:** Always check your work by evaluating derivatives carefully, especially when applying the quotient rule.

Dada la curva paramétrica, determinar la pendiente de la recta tangente cuando t = 0.

Learn how to calculate the slope of the tangent line to the parametric curve at t = 0. This problem involves parametric equations, the quotient rule, and derivatives, suitable for advanced high school or introductory university calculus students.

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