Limit of an Improper Integral

Evaluate $\lim_{{B \to \infty}} \displaystyle \int_{{1}}^{{B}} \frac{1}{x} \, dx$.

This problem involves evaluating a limit of an improper integral, which is a critical concept in calculus and analysis. Improper integrals extend the idea of a definite integral to cases where the interval of integration is unbounded or the integrand becomes infinite within the interval. In this particular problem, the integral of 1 over x from 1 to B is considered as B approaches infinity. This setup helps highlight the behavior of integrals with infinite intervals and their convergence or divergence.

Understanding improper integrals requires the student to grasp both the concept of limits and the fundamental theorem of calculus. This problem challenges the student to analyze the convergence of the integral by examining its antiderivative. The natural logarithm function emerges in this context, showing how logarithmic functions can describe growth and how they potentially diverge as the variable approaches infinity.

Further, this exercise underscores the importance of recognizing when an integral diverges. The harmonic series is an example closely related to this concept, as it also deals with the divergence of sums over an infinite domain. Mastery of improper integrals and limits is crucial for advancing in calculus, especially for understanding series and multivariable calculus concepts later on.

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