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The image asks which of the following limits can help solve the expression:

\[
\lim_{{t \to \infty}} \left[ e^{-0.3t}(t + 2) \right]
\]

There are three options, and I will analyze each to see which one aids in solving this limit:

1. **First option:**
   \[
   \lim_{{t \to \infty}} \left[ t e^{-0.3t} + 2 e^{-0.3t} \right] = \lim_{{t \to \infty}} \left[ t e^{-0.3t} \right] + \lim_{{t \to \infty}} \left[ 2 e^{-0.3t} \right] = 0 + 0 = 0
   \]
   This expression shows the limit of two terms: \(t e^{-0.3t}\) and \(2 e^{-0.3t}\), both of which tend to 0 as \(t \to \infty\) due to the exponential decay overpowering the linear term.

2. **Second option:**
   \[
   \lim_{{t \to \infty}} \left[ e^{-0.3t} (t + 2) \right] = 0 \cdot \infty = 0
   \]
   This option directly solves the original problem. As \(t \to \infty\), \(e^{-0.3t} \to 0\), but because \(t + 2 \to \infty\), this is an indeterminate form (0 times infinity). However, the exponential decay dominates, leading to a result of 0.

3. **Third option:**
   \[
   \lim_{{t \to \infty}} \left[ t e^{-0.3t} \right] + \lim_{{t \to \infty}} \left[ 2 e^{-0.3t} \right] = \infty + 0 = \infty
   \]
   This expression incorrectly claims that the limit is \(\infty\). We know that both terms tend to 0, not \(\infty\), so this is not the correct calculation.

### Conclusion:
The second option provides the correct analysis to solve the limit:
\[
\lim_{{t \to \infty}} \left[ e^{-0.3t}(t + 2) \right] = 0
\]

Would you like a deeper explanation, or do you have any further questions? 

Here are five related questions you might find useful:
1. Why does \(e^{-0.3t}\) dominate \(t\) as \(t \to \infty\)?
2. What happens in other cases where the exponential term decays slower?
3. How does L'Hopital's rule apply to limits involving indeterminate forms like \(0 \times \infty\)?
4. What is the general behavior of terms involving both exponential and polynomial functions in limits?
5. How does this limit change if the exponent on \(e\) is positive?

**Tip:** When dealing with limits involving terms like \(e^{-kt} \cdot f(t)\) where \(f(t)\) grows without bound, check if the exponential decay is stronger to avoid misinterpreting the indeterminate form.

¿Cuál de los siguientes cálculos permite resolver lim_{t \to \infty} \left[ e^{-0.3t}(t + 2) \right]?

Learn how to solve limit problems involving exponential decay, such as lim_{t \to \infty} [e^{-0.3t}(t + 2)]. This problem explores key concepts in calculus, including exponential functions and their limits as t approaches infinity.

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