Limits and Continuity

On what intervals are the following functions continuous?

c. h(x) = cos x / x―π

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Identify the type of function: The function h(x) = cos(x) / (x - π) is a rational function, which is generally continuous everywhere except where the denominator is zero.

Determine where the denominator is zero: Set the denominator equal to zero to find the points of discontinuity. Solve the equation x - π = 0, which gives x = π.

Analyze the continuity: Since the function is a rational function, it is continuous on its domain, which is all real numbers except where the denominator is zero.

Express the intervals of continuity: The function is continuous on the intervals (-∞, π) and (π, ∞), as these intervals do not include the point where the denominator is zero.

Conclude the analysis: Therefore, the function h(x) = cos(x) / (x - π) is continuous on the intervals (-∞, π) and (π, ∞).

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Continuity of Functions

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 crucial for determining where a function does not have breaks, jumps, or asymptotes.

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Limits describe the behavior of a function as it approaches a specific point from either side. Understanding limits is essential for analyzing continuity, especially at points where the function may not be explicitly defined, such as points of division by zero. Evaluating limits helps identify potential discontinuities in the function.

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Piecewise functions are defined by different expressions over different intervals. When analyzing continuity for such functions, it is important to check the limits and values at the boundaries of these intervals. This concept is relevant for determining the overall continuity of functions that may behave differently in separate segments.

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