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


a. ƒ(x) = tan x

Verified step by step guidance

1. Identify the function: The function given is \( f(x) = \tan x \). The tangent function is defined as \( \tan x = \frac{\sin x}{\cos x} \).
2. Determine where the function is undefined: The function \( \tan x \) is undefined where \( \cos x = 0 \) because division by zero is undefined.
3. Find the values of \( x \) where \( \cos x = 0 \): The cosine function is zero at odd multiples of \( \frac{\pi}{2} \), i.e., \( x = \frac{\pi}{2} + k\pi \) where \( k \) is an integer.
4. Identify intervals of continuity: The function \( \tan x \) is continuous on intervals where it is defined, which are the open intervals between the points where \( \cos x = 0 \).
5. Conclude the intervals of continuity: Therefore, \( f(x) = \tan x \) is continuous on intervals of the form \( (k\pi - \frac{\pi}{2}, k\pi + \frac{\pi}{2}) \) for each integer \( k \).

Key Concepts

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 tan(x) as x approaches π/2 is undefined, indicating a vertical asymptote.

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. Discontinuities can occur due to vertical asymptotes, jumps, or holes in the graph, which are essential to identify when determining intervals of continuity.

The Tangent Function

The tangent function, defined as tan(x) = sin(x)/cos(x), is periodic and has vertical asymptotes where the cosine function equals zero, specifically at odd multiples of π/2. This periodicity and the locations of its discontinuities are critical for determining the intervals of continuity for tan(x). Thus, tan(x) is continuous on intervals that do not include these asymptotes.