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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1:30 minutes

Problem 2.6.1h

Textbook Question

Finding Limits

For the function f whose graph is given, determine the following limits. Write ∞ or −∞ where appropriate.

h. lim x → ∞ f(x)

Verified step by step guidance

Observe the graph of the function f(x) as x approaches infinity. This means we are interested in the behavior of the function as x moves towards the right on the x-axis.

Identify the trend of the function f(x) for large values of x. Notice that as x increases, the graph of the function appears to level off and approach a horizontal line.

Determine the y-value that the function is approaching as x goes to infinity. This is the horizontal asymptote of the function.

From the graph, it appears that the function f(x) approaches a constant value as x becomes very large. This constant value is the limit of the function as x approaches infinity.

Conclude that the limit of f(x) as x approaches infinity is the y-value of the horizontal asymptote observed in the graph.

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Key Concepts

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

Limits at Infinity

Limits at infinity describe the behavior of a function as the input approaches positive or negative infinity. This concept helps determine whether the function approaches a specific value, diverges to infinity, or oscillates. Understanding limits at infinity is crucial for analyzing the end behavior of functions, particularly rational functions and those with asymptotes.

Vertical Asymptotes

Vertical asymptotes occur when the function approaches infinity or negative infinity as the input approaches a certain value. This typically happens at points where the function is undefined, such as division by zero. Identifying vertical asymptotes is essential for understanding the overall shape of the graph and the limits of the function near those points.

Continuous Functions

A function is continuous if there are no breaks, jumps, or holes in its graph. For limits, continuity ensures that the limit of the function as it approaches a point equals the function's value at that point. Understanding continuity is vital for evaluating limits, especially when determining the behavior of functions at specific points or as they approach infinity.

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Related Practice

Textbook Question

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→∞ (x − 3) / √(4x² + 25)

Textbook Question

Theory and Examples

Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.

Textbook Question

Theory and Examples

Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

Textbook Question

Formal Definitions of One-Sided Limits

Greatest integer function Find (a) limx→400+ ⌊x⌋ and (b) limx→400− ⌊x⌋; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx→400 ⌊x⌋? Give reasons for your answer.

Textbook Question

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

a. How does the graph behave as x → 0⁺?

Give reasons for your answers.

y = (3/2)(x − (1 / x))²/³

Textbook Question

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

c. How does the graph behave near x = 1 and x = −1?

Give reasons for your answers.

y = (3/2)(x − (1 / x))²/³

Textbook Question

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.