Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.
lim x→π/2^+ tan x

Verified step by step guidance

1. Step 1: Understand the behavior of the tangent function. The function \( y = \tan x \) has vertical asymptotes where the cosine of \( x \) is zero, which occurs at \( x = \frac{\pi}{2} + k\pi \) for any integer \( k \).
2. Step 2: Identify the limit direction. The limit \( \lim_{x \to \frac{\pi}{2}^+} \tan x \) means we are approaching \( \frac{\pi}{2} \) from the right side, or from values slightly greater than \( \frac{\pi}{2} \).
3. Step 3: Analyze the behavior of \( \tan x \) as \( x \to \frac{\pi}{2}^+ \). As \( x \) approaches \( \frac{\pi}{2} \) from the right, \( \tan x \) tends to increase without bound because the cosine of \( x \) approaches zero from the positive side, making \( \tan x = \frac{\sin x}{\cos x} \) approach positive infinity.
4. Step 4: Sketch the graph of \( y = \tan x \) over the interval \([-\pi, \pi]\). Note the vertical asymptotes at \( x = -\frac{\pi}{2} \) and \( x = \frac{\pi}{2} \), and the periodic nature of the tangent function with period \( \pi \).
5. Step 5: Use the graph to verify the limit. On the graph, observe that as \( x \) approaches \( \frac{\pi}{2} \) from the right, the value of \( \tan x \) increases towards positive infinity, confirming the limit analysis.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of the tangent function as x approaches π/2 from the right. Understanding limits helps in analyzing the continuity and behavior of functions, especially at points where they may not be defined.

Tangent Function

The tangent function, denoted as tan(x), is a periodic function defined as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). It has vertical asymptotes where the cosine function is zero, such as at x = π/2, leading to undefined values. Recognizing the properties of the tangent function is crucial for analyzing its limits and graphing its behavior.

Graphing and Asymptotes

Graphing functions involves plotting their values on a coordinate system to visualize their behavior. For the tangent function, it is essential to identify vertical asymptotes, which occur at points where the function approaches infinity. In this case, as x approaches π/2 from the right, the graph of y = tan(x) will rise steeply, illustrating the concept of limits and the function's undefined nature at that point.