Textbook Question
A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.
lim x→−2^− f(x)
1. Limits and Continuity
Introduction to Limits
1:38 minutesProblem 51c
Textbook Question
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 guidance1
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.
Step 2: Consider the limit \( \lim_{x \to \frac{\pi}{2}^+} \tan x \). This means we are approaching \( \frac{\pi}{2} \) from the right side (values greater than \( \frac{\pi}{2} \)).
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.
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.
Step 5: Use the graph to verify that as \( x \to \frac{\pi}{2}^+ \), \( \tan x \to +\infty \). This confirms the behavior of the limit as described in the previous steps.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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