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.