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We are given the integral:

\[
\int_0^3 \frac{e^{-2/t^3}}{t^4} \, dt
\]

and we need to determine whether this improper integral converges or diverges. To do so, we will analyze the behavior at both \( t = 0 \) and \( t = 3 \), particularly since \( t = 0 \) may lead to singularities due to the \( t^4 \) in the denominator.

### Steps:

1. **Near \( t = 3 \)**: The function \( \frac{e^{-2/t^3}}{t^4} \) does not pose any issues at \( t = 3 \) because the exponential decays very rapidly, making the function behave regularly. So, no need to worry about divergence near \( t = 3 \).

2. **Near \( t = 0 \)**: This is the critical point because the denominator \( t^4 \) tends to infinity as \( t \to 0^+ \), and we need to check how the integrand behaves near this singularity.

To rigorously check this, we would split the integral near 0:

\[
\int_0^3 \frac{e^{-2/t^3}}{t^4} dt = \lim_{a \to 0^+} \int_a^3 \frac{e^{-2/t^3}}{t^4} dt
\]

Now, let’s compute the behavior as \( t \to 0 \).

The key thing to analyze here is how fast \( e^{-2/t^3} \) decays. Near 0, the exponential term \( e^{-2/t^3} \) decays extremely fast (faster than any polynomial growth of \( 1/t^4 \)). This rapid decay of the exponential factor ensures that the integral near \( t = 0 \) converges.

### Conclusion:

The integral converges because the exponential term decays sufficiently fast near \( t = 0 \), overpowering the singularity from \( 1/t^4 \).

Would you like more detailed calculations on this or have any specific questions?

Here are five related questions to extend this concept:
1. How does the exponential decay affect the convergence of integrals in general?
2. What are common techniques to test convergence of improper integrals?
3. How would the result change if the power of \( t \) in the denominator were different, e.g., \( t^5 \)?
4. What if we changed the limits of the integral to infinity?
5. Can improper integrals have different behaviors based on the location of singularities?

**Tip**: When dealing with improper integrals, always focus on points where the integrand may become infinite or undefined and analyze the limiting behavior carefully.

Determine if the following integral converges or diverges: ∫ (0 to 3) (e^(-2/t^3))/(t^4) dt.

This problem explores the convergence or divergence of the improper integral ∫ (0 to 3) (e^(-2/t^3))/(t^4) dt. The solution investigates the behavior of the integrand near potential singularities, particularly at t = 0, and applies key techniques in analyzing improper integrals.

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