Determine whether the following statements are true and give an explanation or counterexample. e. lim x → π 2 cot x = 0 {\displaystyle\lim_{x\to\frac{\pi}{2}}}\cot x=0 . (Hint: Graph y=cot x)

Welcome back, everyone. Determine the validity of the given statement. Limit of cotangent X as X approaches 3 pi divided by 2 is equal to 0. We're given to answer choices A says true and B says false. Whenever we want to evaluate limits, the first step is to use direct substitution, so we can simply substitute x equals 3 pi divided by 2 into the function, and we are going to check if it is continuous at a given point. If it is, well, we're basically done. So limit as x approaches 3 pi divided by 2 of cotangent X. It's going to be equal to, now we are checking if this function is continuous at a given point, so we're taking cotangent at 3 pi divided by 2. We can evaluate this result. Which essentially gives us 0. It is a finite number. We don't have any indeterminate forms, and this essentially means that we have successfully evaluated the limit. The given statement says that the limit is equal to 0, which is consistent with the calculation. So we can see that the correct answer to this problem is a true. Thank you for watching.

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
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7. Antiderivatives & Indefinite Integrals1h 26m
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- Introduction to Power Series
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16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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Problem 33e

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

e. $\lim_{x \to \frac{π}{2}} \cot x = 0$ . (Hint: Graph y=cot x)

Verified step by step guidance

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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Key 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.

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.

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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Determine whether the following statements are true and give an explanation or counterexample.e. limx→π2cotx=0{\displaystyle\lim_{x\to\frac{\pi}{2}}}\cot x=0 . (Hint: Graph y=cot x)

Key Concepts

Limit of a Function

Cotangent Function

Graphical Analysis

Watch next

Determine whether the following statements are true and give an explanation or counterexample.

e. $\lim_{x \to \frac{π}{2}} \cot x = 0$ . (Hint: Graph y=cot x)