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 left side, or from values less than \( \frac{\pi}{2} \).
3. Step 3: Analyze the behavior of \( \tan x \) as \( x \) approaches \( \frac{\pi}{2} \) from the left. As \( x \) approaches \( \frac{\pi}{2}^- \), \( \tan x \) increases 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 left, the value of \( \tan x \) increases towards positive infinity, confirming the limit analysis.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this case, we are interested in the limit of the tangent function as x approaches π/2 from the left. Understanding limits helps in analyzing the behavior of functions near points of discontinuity or asymptotes.

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, such as at x = π/2. This characteristic leads to the function approaching infinity as x approaches these points from the left or right, which is crucial for evaluating the limit in the question.

Graphing Functions

Graphing functions provides a visual representation of their behavior, including limits and asymptotes. By sketching the graph of y = tan(x) over the specified window, one can observe the function's approach to infinity as x nears π/2, confirming the analytical limit found. This visual check enhances understanding of the function's properties.