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, $y=\tan x$, near $x=\frac{\pi }{2}$. The tangent function is undefined at $x=\frac{\pi }{2}$ because it corresponds to a vertical asymptote.
2. Step 2: Consider the limit $\underset{x\to {\frac{\pi }{2}}^{+}}{lim}\tan x$. This means we are approaching $\frac{\pi }{2}$ from the right side (values greater than $\frac{\pi }{2}$).
3. Step 3: As $x$ approaches $\frac{\pi }{2}$ from the right, the values of $\tan x$ increase without bound. This is because the tangent function approaches positive infinity as it nears the vertical asymptote from the right.
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 observe the behavior of the function as it approaches these asymptotes.
5. Step 5: Use the graph to verify that as $x\to {\frac{\pi }{2}}^{+}$, $\tan x\to +\mathrm{\infty }$. This confirms the behavior of the limit as described in the previous steps.

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 sketching its graph.

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize the behavior of the function. For the tangent function, understanding its periodic nature and asymptotes is essential for accurate representation. By sketching the graph of y = tan(x) over the specified window, one can visually confirm the behavior of the function near the limit, enhancing comprehension of the limit's value.