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


d. k(x) = sin x / x

Verified step by step guidance

1. First, recall the definition of continuity: a function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.
2. Consider the function k(x) = sin(x) / x. This function is defined for all x except x = 0, because division by zero is undefined.
3. To determine the intervals of continuity, we need to check where the function is defined and where it is continuous. Since k(x) is undefined at x = 0, it cannot be continuous there.
4. For x ≠ 0, the function is continuous because both sin(x) and x are continuous functions, and the quotient of two continuous functions is continuous wherever the denominator is non-zero.
5. Therefore, the function k(x) = sin(x) / x is continuous on the intervals (-∞, 0) and (0, ∞).

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are crucial for understanding continuity, as a function is continuous at a point if the limit exists and equals the function's value at that point. For the function k(x) = sin x / x, evaluating the limit as x approaches 0 is essential to determine its continuity.

Continuity

A function is continuous at a point if three conditions are met: the function is defined at that point, the limit exists at that point, and the limit equals the function's value. For k(x) = sin x / x, we need to check its behavior at x = 0 and other points to determine where it is continuous. Understanding the definition of continuity helps in identifying intervals of continuity for various functions.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. For k(x) = sin x / x, it is important to recognize that while the function is not defined at x = 0, it can be extended to be continuous by defining k(0) = 1, which is the limit of k(x) as x approaches 0. This concept is vital for analyzing functions that may have discontinuities at specific points.