Limits and Continuity


On what intervals are the following functions continuous?


c. h(x) = cos x / x―π

Verified step by step guidance

1. First, recall that a function is continuous at a point if the limit of the function as it approaches that point is equal to the function's value at that point. For a function to be continuous on an interval, it must be continuous at every point within that interval.
2. Consider the function h(x) = \( \frac{\cos x}{x - \pi} \). The function is defined for all x except where the denominator is zero, which occurs at x = \pi. Therefore, h(x) is not defined at x = \pi, and hence not continuous there.
3. Next, examine the behavior of h(x) as x approaches \pi. Since the denominator becomes zero at x = \pi, we need to check if the limit exists as x approaches \pi from both sides. This involves evaluating the limit \( \lim_{x \to \pi} \frac{\cos x}{x - \pi} \).
4. To determine the intervals of continuity, note that h(x) is continuous wherever it is defined and the limit exists. Since the only point of discontinuity is at x = \pi, h(x) is continuous on the intervals \((-\infty, \pi)\) and \((\pi, \infty)\).
5. Finally, conclude that the function h(x) = \( \frac{\cos x}{x - \pi} \) is continuous on the intervals \((-\infty, \pi)\) and \((\pi, \infty)\), excluding the point x = \pi where it is not defined.