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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1:25 minutes

Problem 82a

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^−1

Verified step by step guidance

Identify the function: The function given is $ y = \cot^{-1}(x) $, which is the inverse cotangent function.

Understand the behavior of $ \cot^{-1}(x) $: As $ x \to \infty $, the inverse cotangent function approaches a specific value.

Recall the range of $ \cot^{-1}(x) $: The range of $ \cot^{-1}(x) $ is $(0, \pi)$.

Analyze the limit: As $ x \to \infty $, the value of $ \cot^{-1}(x) $ approaches the lower bound of its range.

Conclude the limit: Based on the behavior of the inverse cotangent function, determine the limit as $ x \to \infty $.

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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 and limits, especially as x approaches infinity.

Limits in Calculus

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. In this context, evaluating the limit of cot^−1(x) as x approaches infinity helps determine the horizontal asymptote of the function, which is essential for understanding its long-term behavior.

Graphical Interpretation of Limits

Graphical interpretation of limits involves analyzing the graph of a function to determine its behavior as the input approaches a specific value. For cot^−1(x), examining the graph as x approaches infinity allows us to visually assess the limit and understand how the function behaves at extreme values.

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Textbook Question

If a function f represents a system that varies in time, the existence of lim $\lim_{t \to \infty} f (t)$ 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.

The population of a bacteria culture is given by $p (t) = \frac{2500}{t + 1}$ .

Textbook Question

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.

The population of a culture of tumor cells is given by $p (t) = \frac{3500 t}{t + 1}$ .

Textbook Question

The population of a colony of squirrels is given by $p (t) = \frac{1500}{3 + 2 e^{- 0.1 t}}$ .

Textbook Question

The hyperbolic cosine function, denoted $\cosh\left(x\right)$ , is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\cosh\left(x\right)=\frac{e^{x}+e^{-x}}{2}$ .