Textbook Question
Use a graph of f to estimate or to show that the limit does not exist. Evaluate f(x) near to support your conjecture.
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1. Limits and Continuity
Introduction to Limits
1:26 minutesProblem 33e
Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
e. . (Hint: Graph y=cot x)
Verified step by step guidance1
insert step 1: Recall the definition of the cotangent function, which is \( \cot x = \frac{\cos x}{\sin x} \).
insert step 2: Consider the behavior of \( \sin x \) and \( \cos x \) as \( x \to \frac{\pi}{2} \). Note that \( \sin \left( \frac{\pi}{2} \right) = 1 \) and \( \cos \left( \frac{\pi}{2} \right) = 0 \).
insert step 3: Substitute these values into the cotangent function: \( \cot x = \frac{\cos x}{\sin x} \to \frac{0}{1} = 0 \).
insert step 4: However, consider the limit \( \lim_{x \to \frac{\pi}{2}} \cot x \). As \( x \) approaches \( \frac{\pi}{2} \) from the left, \( \cos x \) approaches 0, but \( \sin x \) is positive and close to 1, making \( \cot x \) approach 0.
insert step 5: As \( x \) approaches \( \frac{\pi}{2} \) from the right, \( \cos x \) is still approaching 0, but \( \sin x \) is negative and close to -1, making \( \cot x \) approach 0. Thus, the limit does not exist as it approaches different values from the left and right.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Function
The limit of a function describes the behavior of that function as the input approaches a certain value. In this case, we are interested in the limit of the cotangent function as x approaches π/2. Understanding limits is crucial for analyzing the continuity and behavior of functions at specific points, especially where they may not be defined.
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Cotangent Function
The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). It is important to note that cot(x) is undefined at points where sin(x) = 0, such as x = nπ, where n is an integer. This characteristic affects the limit as x approaches π/2, where the function exhibits vertical asymptotic behavior.
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Graphical Analysis
Graphical analysis involves examining the graph of a function to understand its behavior visually. For the cotangent function, plotting y = cot(x) reveals that as x approaches π/2, the function tends toward negative infinity, not zero. This visual representation helps clarify the limit's value and provides insight into the function's discontinuities and asymptotes.
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