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

d. k(x) = x⁻¹/⁶

검증된 단계별 안내

Step 1: Understand the function k(x) = x^-1/6. This function is a power function where the exponent is negative, indicating that it involves a root in the denominator.

Step 2: Recall the definition of continuity. A function is continuous at a point if it is defined at that point, the limit exists at that point, and the limit equals the function value.

Step 3: Identify the domain of k(x). Since k(x) = x^-1/6 involves a root in the denominator, it is undefined for x = 0. Therefore, the function is not continuous at x = 0.

Step 4: Consider the behavior of the function for x > 0 and x < 0. For x > 0, the function is defined and continuous because the root is real and positive. For x < 0, the function is also defined and continuous because the root is real and negative.

Step 5: Conclude the intervals of continuity. The function k(x) = x^-1/6 is continuous on the intervals (-∞, 0) and (0, ∞), excluding x = 0 where the function is undefined.

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주요 개념

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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Understanding limits is crucial for analyzing the behavior of functions, especially at points where they may not be explicitly defined. For example, the limit of k(x) as x approaches 0 helps determine the continuity of the function at that point.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. This concept is essential for determining where the function k(x) = x⁻¹/⁶ is continuous, particularly around points where the function may be undefined.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For k(x) = x⁻¹/⁶, the function is undefined when x = 0, as it would involve division by zero. Identifying the domain is critical for determining the intervals of continuity, as the function can only be continuous where it is defined.

Limits and ContinuityOn what intervals are the following functions continuous?d. k(x) = x⁻¹/⁶