Table of contents

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
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
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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1:54 minutes

Problem 82b

Textbook Question

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.
lim x→−∞ cot^−1x

Verified step by step guidance

Step 1: Understand the function $ y = \cot^{-1}(x) $. The inverse cotangent function, $ \cot^{-1}(x) $, is the angle whose cotangent is $ x $. It is defined for all real numbers and its range is $(0, \pi)$.

Step 2: Consider the behavior of $ \cot^{-1}(x) $ as $ x \to -\infty $. As $ x $ becomes very large in the negative direction, the angle whose cotangent is $ x $ approaches $ \pi $.

Step 3: Visualize the graph of $ y = \cot^{-1}(x) $. As $ x \to -\infty $, the graph approaches the horizontal asymptote at $ y = \pi $.

Step 4: Use the graph to determine the limit. The graph shows that as $ x \to -\infty $, $ \cot^{-1}(x) $ approaches $ \pi $.

Step 5: Conclude that $ \lim_{x \to -\infty} \cot^{-1}(x) = \pi $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Cotangent Function

The inverse cotangent function, denoted as cot^−1(x), is the function that returns the angle whose cotangent is x. It is defined for all real numbers and has a range of (0, π). Understanding this function is crucial for analyzing its behavior as x approaches different limits.

Limits in Calculus

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. Evaluating limits helps in understanding the behavior of functions at boundaries, including infinity. In this case, we are interested in the limit of cot^−1(x) as x approaches negative infinity.

Behavior of Functions at Infinity

The behavior of functions as they approach infinity or negative infinity is essential for understanding their long-term trends. For the cotangent function, as x approaches negative infinity, the value of cot^−1(x) approaches π. This concept helps in predicting the output of the function without direct computation.

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If a function f represents a system that varies in time, the existence of lim ${\displaystyle\lim_{t\rightarrow\infty}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

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