# How do you determine if the improper integral converges or diverges #int [e^(1/x)] / [x^3]# from 0 to 1?

The integral diverges.

First focusing on the integral without bounds:

Performing integration by parts, letting:

So:

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To determine if the improper integral ( \int_{0}^{1} \frac{e^{1/x}}{x^3} , dx ) converges or diverges:

- Analyze the behavior of the integrand near the point of integration, ( x = 0 ).
- Determine if the integrand approaches zero fast enough as ( x ) approaches zero.
- Check if the integral ( \int_{0}^{1} \frac{e^{1/x}}{x^3} , dx ) can be transformed into a known convergent or divergent integral.

Now, let's proceed with the steps:

- Near ( x = 0 ), the function ( e^{1/x} ) grows rapidly as ( x ) approaches zero, while ( x^3 ) approaches zero.
- We can use the Limit Comparison Test to compare the given integral with a known integral to determine convergence or divergence.
- Let's consider the function ( f(x) = \frac{e^{1/x}}{x^3} ). As ( x ) approaches zero, ( e^{1/x} ) grows much faster than ( x^3 ).
- We'll compare ( f(x) ) with ( g(x) = \frac{1}{x^3} ), which is easier to integrate: [ \lim_{x \to 0^+} \frac{f(x)}{g(x)} = \lim_{x \to 0^+} \frac{e^{1/x}/x^3}{1/x^3} = \lim_{x \to 0^+} e^{1/x} = +\infty ]
- Since the limit is infinity, the integral ( \int_{0}^{1} \frac{e^{1/x}}{x^3} , dx ) diverges.

Therefore, the improper integral ( \int_{0}^{1} \frac{e^{1/x}}{x^3} , dx ) diverges.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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