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The problem is asking for the torsion \( \tau(t) \) of a space curve defined by the vector function \( \vec{f}(t) \) in \( \mathbb{R}^3 \), given specific expressions for the magnitude of the velocity vector \( ||\vec{f}'(t)|| \) and the derivative of the binormal vector \( \vec{B}'(t) \). Let's go through the provided information and interpret the problem step by step.

### Key Information:
- The magnitude of the derivative of \( \vec{f}(t) \) (the speed) is given as:
  \[
  ||\vec{f}'(t)|| = \frac{1}{t + 1}
  \]
- The derivative of the binormal vector \( \vec{B}(t) \) is given as:
  \[
  \vec{B}'(t) = \frac{1}{(1 + t)^2}(-1, -1, \frac{1 - t}{\sqrt{2t}})
  \]
- We are tasked with finding the torsion \( \tau(t) \) as \( t \to \infty \).

### Torsion Formula:
The torsion \( \tau(t) \) of a curve in \( \mathbb{R}^3 \) is defined as:
\[
\tau(t) = -\frac{\vec{B}'(t) \cdot \vec{T}(t)}{||\vec{f}'(t)||}
\]
Where:
- \( \vec{B}(t) \) is the binormal vector.
- \( \vec{T}(t) \) is the tangent vector \( \vec{f}'(t) / ||\vec{f}'(t)|| \).
- \( \vec{B}'(t) \) is the derivative of the binormal vector.

Given that \( t \to \infty \), we need to analyze the behavior of \( \vec{B}'(t) \) and \( ||\vec{f}'(t)|| \) under that limit.

### Behavior as \( t \to \infty \):
1. \( ||\vec{f}'(t)|| = \frac{1}{t + 1} \), so as \( t \to \infty \), this tends to 0.
2. For \( \vec{B}'(t) \), observe the terms in the vector. As \( t \to \infty \), each component behaves as follows:
   - \( \frac{1}{(1+t)^2} \to 0 \).
   - The components \( (-1, -1, \frac{1 - t}{\sqrt{2t}}) \) will scale accordingly.

Now, combining these results and focusing on how the torsion \( \tau(t) \) behaves as \( t \to \infty \), we can deduce that the torsion will tend towards **0** because both \( \vec{B}'(t) \) and \( ||\vec{f}'(t)|| \) tend to zero at a similar rate.

Thus, the correct answer is **B) 0**.

### Would you like further details or have any questions?

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Here are 5 related questions for further exploration:
1. What is the formula for torsion in terms of the Frenet-Serret equations?
2. How does curvature \( \kappa(t) \) differ from torsion in describing space curves?
3. How would the result change if the curve was in \( \mathbb{R}^2 \)?
4. How does the binormal vector \( \vec{B}(t) \) evolve for different space curves?
5. How would you compute the curvature \( \kappa(t) \) for this problem?

**Tip:** Torsion measures how much a curve "twists" in space, while curvature describes how much it bends.

Sea C una curva en R^3, definida por la función vectorial f(t), tales que ||f'(t)|| = 1/(t+1) y B'(t) = 1/(1+t)^2 (-1, -1, (1-t)/sqrt(2t)), para t > 0. La torsión τ(t) a medida que t crece es:

Learn how to calculate the torsion of a curve in 3D space defined by vector functions. Explore the Frenet-Serret formulas, and use key mathematical concepts such as vector calculus and torsion. Suitable for university-level students studying advanced calculus or differential geometry.

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