Question

Graph the function \(y=\tan x\) with the window \([-\pi, \pi] \times[-10,10] .\) Use the graph to analyze the following limits. a. \(\lim _{x \rightarrow \pi / 2^{+}} \tan x\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \tan x\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x\)

Short answer

In summary, the limits of the tangent function y = tan(x) within the given interval are: a. \(\lim _{x \rightarrow \pi / 2^{+}} \tan x = +\infty\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \tan x = -\infty\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x = -\infty\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x = +\infty\) This shows that the tangent function has vertical asymptotes at x = ±π/2 within the given interval, and its behavior is different when approaching these points from the left or right.

Step-by-step solution

Graph the function

To graph the function, we'll first need to find the range for the tangent function between \([-\pi,\pi]\). We will also be confined to \(-10<y<10\). The tangent function has vertical asymptotes at x = \(\pm\frac{\pi}{2}\), \(\pm\frac{3\pi}{2}\), etc. Within the given interval \([-\pi, \pi]\), we have vertical asymptotes at \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). Now, graph the function within these bounds.

Analyze the limits

Now, analyze the graph to deduce the limits: a. \(\lim _{x \rightarrow \pi / 2^{+}} \tan x\) When \(x\) approaches \(\pi / 2\) from the right (positive side), the value of the tangent function increases without bound. This means: \(\lim _{x \rightarrow \pi / 2^{+}} \tan x = +\infty\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \tan x\) When \(x\) approaches \(\pi / 2\) from the left (negative side), the value of the tangent function decreases without bound. This means: \(\lim _{x \rightarrow \pi / 2^{-}} \tan x = -\infty\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x\) When \(x\) approaches \(-\pi / 2\) from the right (positive side), the value of the tangent function decreases without bound. This means: \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x = -\infty\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x\) When \(x\) approaches \(-\pi / 2\) from the left (negative side), the value of the tangent function increases without bound. This means: \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x = +\infty\)

Key concepts

Limits of Trigonometric Functions

Understanding the limits of trigonometric functions is essential for analyzing the behavior of these functions around specific points. A limit describes the value that a function approaches as the input (or 'x' value) approaches a certain point. For trigonometric functions like the tangent, sine, and cosine, these limits can be finite or infinite and are crucial for comprehending their characteristics.

For the tangent function, which is periodic and has undefined values at certain points (where the cosine function is zero), these limits can often lead to infinite values, which signify that the function grows without bound in either the positive or negative direction as it approaches certain x-values.

Vertical Asymptotes

A vertical asymptote is a vertical line that a function approaches but never touches or crosses. These lines represent points on the graph where the function's value becomes infinitely large or small—that is, the function is undefined. In terms of the graph of the tangent function, vertical asymptotes occur at values of x where the cosine function is zero, resulting in an undefined value for the tangent (since the tangent function is the sine function divided by the cosine function).

When graphing functions, recognizing these asymptotes is critical for a correct representation of the function’s behavior around these points. For the tangent function, these occur at x = ±(π/2) + kπ, where k is an integer. These points lead to behaviors such as where the function abruptly switches from positive to negative infinity (or vice versa), demonstrating the strikingly different nature of limits from either side of the asymptote.

Infinite Limits

In the realm of calculus, an infinite limit refers to a situation where a function increases or decreases without bound near a certain point along the x-axis, resulting in the function heading towards positive or negative infinity. With the tangent function, this occurs as the x-values approach the vertical asymptotes.

In the given exercise, the tangent function exhibits infinite limits as x approaches ±(π/2). These types of limits are crucial for disciplines such as engineering and physics because they can signal a system's points of instability, where small changes in input lead to large, uncontrolled changes in output.

Tangent Function Behavior

The tangent function behavior is unique in several ways. Its graph comprises repeating patterns, known as periods, and within each period, the function has both a minimum and maximum turning point but no maximum or minimum value since it extends infinitely.

The behavior near the vertical asymptotes is of particular interest. As seen in the exercise solutions, the function approaches different infinities depending on the direction from which x approaches the asymptote. For instance, as x approaches π/2 from the right, the tangent function tends towards positive infinity; approaching from the left, it goes towards negative infinity. This concept reflects in the given limits where the tangent function’s infinite behavior is captured by the limits being equal to +∞ or -∞, depending on whether x approaches the asymptote from the positive or negative side.

Observing these tendencies allows students to better understand the overall structure and behavior of trigonometric functions, paving the way for a comprehensive grasp of their properties.