Evaluate the Limit of a Trigonometric Integral

Evaluate $\displaystyle \lim_{{B \to \infty}} \int_{{0}}^{{B}} \cos x \, dx$.

When approaching the evaluation of an integral where the limit of the upper bound extends to infinity, we are dealing with an improper integral. In this context, considering the limit of the integral as the upper bound approaches infinity, requires us to first determine the indefinite integral and then analyze the behavior of the resulting expression as the boundary extends indefinitely. In this problem, the integrand involves the cosine function which is a periodic function. This specific characteristic of trigonometric functions plays a key role in analyzing the convergence or divergence of such limits. For cosine, the integral over a full period is zero, which gives a hint about the behavior of the integral as the upper limit approaches infinity. Evaluating the improper integral requires understanding both the properties of the cosine function and the concept of limits in calculus. Keep in mind that sometimes, even though an indefinite integral can be computed, it might not lead straightforwardly to a solution if the resulting expression does not converge to a finite number within the bounds given in the limit. Understanding the relationship between periodicity and limits is key to solving this type of improper integral problem.

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