We are tasked with determining whether the following integral is convergent or divergent, and if it is convergent, evaluating it:

$\int_{-1}^{5} \frac{e^{1/x}}{x^3} \, dx$

Step 1: Examine the Behavior of the Integrand

We need to analyze the behavior of the integrand $\frac{e^{1/x}}{x^3}$ near potential points where the function may be undefined or where there could be issues with convergence. Specifically, we need to check:

• The behavior near $x = 0$, as this is a point where $x^3$ in the denominator could cause a singularity.
• The limits at $x = -1$ and $x = 5$, although these are finite, it's important to confirm there are no additional issues there.

Step 2: Behavior Near $x = 0$

The function $\frac{e^{1/x}}{x^3}$ has a singularity at $x = 0$, and we need to assess whether this singularity causes the integral to diverge. As $x \to 0^+$, $e^{1/x}$ grows extremely fast (it tends to infinity), and the term $\frac{1}{x^3}$ also blows up. Thus, we anticipate the integral might be divergent around $x = 0$.

Step 3: Check Convergence Using Limits

We can check the convergence by breaking the integral into two parts and evaluating the limits separately.

1. From $-1$ to some small positive value $\epsilon$: $\int_{-1}^{\epsilon} \frac{e^{1/x}}{x^3} \, dx$

2. From some small positive value $\epsilon$ to $5$: $\int_{\epsilon}^{5} \frac{e^{1/x}}{x^3} \, dx$

The second part of the integral will not present issues, but the first part is where we need to focus on $x = 0$.

Step 4: Conclusion

The integrand $\frac{e^{1/x}}{x^3}$ blows up as $x \to 0^+$, and because $e^{1/x}$ grows so rapidly, the integral does not converge in this region. Therefore, the integral diverges.

Final Answer:

$\boxed{\text{DIVERGES}}$

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Would you like a deeper explanation of the specific divergence behavior near $x = 0$, or have any questions about the method used?

Here are some related questions:

1. How do you handle integrals with singularities at specific points?
2. What methods can be used to assess convergence of improper integrals?
3. What happens to integrals where the numerator grows exponentially, like $e^{1/x}$?
4. How do you deal with improper integrals involving powers of $x$?
5. Can you solve similar integrals that involve limits approaching infinity?

Tip: Always carefully analyze the behavior of an integrand near points where it might be undefined, such as at singularities or infinite limits, before proceeding with the integration.