Limits and Continuity
On what intervals are the following functions continuous?


d. k(x) = sin x / x

Verified step by step guidance

1. Identify the type of function: The function k(x) = \( \frac{\sin x}{x} \) is a rational function, which is generally continuous wherever the denominator is not zero.
2. Determine where the function is undefined: The function is undefined at x = 0 because the denominator becomes zero, leading to a division by zero.
3. Consider the continuity of the numerator: The sine function, \( \sin x \), is continuous everywhere on the real number line.
4. Analyze the intervals of continuity: Since the function is undefined at x = 0, it is continuous on the intervals (-∞, 0) and (0, ∞).
5. Conclude the intervals of continuity: The function k(x) = \( \frac{\sin x}{x} \) is continuous on the intervals (-∞, 0) and (0, ∞), but not at x = 0.