Limit of Improper Integral Using Integration by Parts

Evaluate $\lim_{{a \to -\infty}} \int_{{a}}^{{0}} e^x \, dx$ using integration by parts.

In this problem, you are tasked with evaluating the limit of an improper integral using the technique of integration by parts. Improper integrals typically deal with infinite limits of integration or integrands with infinite discontinuities. Evaluating these integrals involves understanding the behavior of the integrand as it approaches the bounds of integration, which can often involve approaching infinity. In this case, we have an integral with one such infinite bound: negative infinity. Integration by parts is a technique derived from the product rule of differentiation. It’s a useful tool for integrals where straightforward integration is complicated. The strategy involves identifying parts of the integrand to differentiate or integrate, thereby simplifying the integral into a more manageable form. For this problem, you will need to cleverly choose which part of the integrand will be differentiated and which part will be integrated. Since the exponential function is involved here, consider how its properties might influence these choices.

The use of limits in this context requires additional understanding of how improper integrals can behave. As you determine the integral from negative infinity to zero, you will need to consider whether the result converges to a finite number or diverges. It’s essential to evaluate this behavior with respect to the properties of exponential functions. This problem provides a comprehensive exercise in both the technique of integration by parts and the concept of limits with improper integrals, allowing you to deepen your understanding of how these mathematical concepts can be used together to solve complex problems.

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