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The given question asks for the limit of a sequence \( u_n \), which is defined as follows:

\[
u_n = \int_0^{+\infty} \frac{dt}{1 + t^2 + n e^{-t}}
\]

We are tasked with finding \( \lim_{n \to +\infty} u_n \).

### Step-by-step solution:

1. **Examine the integrand**: The integrand is:
   \[
   \frac{1}{1 + t^2 + n e^{-t}}
   \]
   As \( n \to +\infty \), the term \( n e^{-t} \) dominates the denominator for large \( n \) because \( e^{-t} \) decreases exponentially as \( t \) increases, but is still significant for small values of \( t \).

2. **Limit for large \( n \)**:
   - For large \( n \), the term \( n e^{-t} \) dominates when \( t \) is not too large, so we approximate the integrand as:
     \[
     \frac{1}{n e^{-t}} = \frac{e^t}{n}
     \]
     This approximation holds for small \( t \), where \( e^{-t} \) is still significant. The integral then behaves as:
     \[
     u_n \sim \int_0^{+\infty} \frac{e^t}{n} dt = \frac{1}{n} \int_0^{+\infty} e^t dt
     \]
     which clearly goes to 0 as \( n \to +\infty \).

3. **Large \( t \) behavior**:
   - As \( t \) becomes large, \( e^{-t} \) decays rapidly, and the denominator becomes dominated by \( 1 + t^2 \). In this case, the integrand behaves like \( \frac{1}{1 + t^2} \), which is integrable over \( [0, +\infty) \) and gives a result of \( \frac{\pi}{2} \).

Thus, the sequence \( u_n \) tends to \( \frac{\pi}{2} \) as \( n \to +\infty \).

### Conclusion:
The correct limit is:
\[
\lim_{n \to +\infty} u_n = \frac{\pi}{2}
\]

The answer is **A**.

---

Would you like a further explanation of any step? Here are 5 related questions to explore this concept further:

1. How does the presence of \( e^{-t} \) affect the behavior of integrals for large \( t \)?
2. Why does \( \frac{1}{1 + t^2} \) result in an integral of \( \frac{\pi}{2} \)?
3. Can this problem be approached using numerical methods for checking the limit?
4. How would the behavior of the sequence change if \( n e^{-t} \) were replaced by a different function?
5. What other mathematical techniques could be used to evaluate limits involving integrals?

**Tip:** When evaluating limits of integrals, look for dominant terms in the integrand as the parameter tends to infinity or zero.

On pose un = ∫₀⁺∞ dt / (1 + t² + n e⁻ᵗ). La limite limₙ→∞ un est :

Solve for the limit of a sequence defined by a definite integral with an asymptotic approximation for large n. Explore key concepts like the dominated convergence theorem and the behavior of integrals at infinity.

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