Evaluating an Improper Integral with Infinite Limit

Evaluate the improper integral $\displaystyle \int_{1}^{\infty} \frac{1}{x^2} \, dx$.

Improper integrals pose unique challenges because they extend the concept of definite integrals to cases where either the interval of integration is infinite or the function to be integrated becomes infinite within the interval. In this problem, you are asked to evaluate an improper integral where the interval extends to infinity. The function within the integral is a simple rational function, which often allows for straightforward evaluation when limits are applied correctly.

The key approach to solving improper integrals with infinite limits is to convert the infinite range into a limit problem. This involves replacing the improper bounds with a variable that approaches infinity, transforming the original improper integral into a limit of a proper integral. For the integral given, you start by considering the integral from 1 to a finite value, say, b, and then analyze the behavior as b approaches infinity. This allows you to apply fundamental integration techniques followed by a limit operation.

Understanding the behavior of the function as the variable approaches infinity or a point of discontinuity is crucial. This process will yield insights into whether the integral converges to a finite value or diverges to infinity. In cases where the antiderivative of the function is easily found, the analysis can often be direct and succinct, making it a good example of handling infinity in calculus.

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