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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2:23 minutes

Problem 2.6.35

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)

Verified step by step guidance

Identify the highest power of x in the denominator. In this case, the highest power is x² under the square root, which simplifies to x when considering the square root.

Divide both the numerator and the denominator by x, the highest power of x in the denominator. This gives us: (x/x - 3/x) / (√(4x² + 25)/x).

Simplify the expression: The numerator becomes (1 - 3/x) and the denominator becomes √(4 + 25/x²).

Evaluate the limit as x approaches infinity: As x → ∞, 3/x approaches 0 and 25/x² approaches 0.

The expression simplifies to (1 - 0) / √(4 + 0), which further simplifies to 1/√4. Calculate the final limit value from this simplified expression.

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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 involve finding the behavior of a function as the variable approaches positive or negative infinity. This concept helps determine the end behavior of functions, particularly rational functions, by analyzing the dominant terms. Understanding limits at infinity is crucial for evaluating how functions grow or shrink without bound.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. To find limits involving rational functions, especially as x approaches infinity, it's essential to identify the highest power of x in the denominator and numerator. This helps simplify the function and determine its limit by focusing on the dominant terms that influence the function's behavior.

Simplification Techniques

Simplification techniques involve dividing the numerator and denominator by the highest power of x present in the denominator. This process reduces the complexity of the function, allowing for easier evaluation of limits. By simplifying, we can isolate terms that significantly impact the limit, making it possible to determine the function's behavior as x approaches 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→∞ (2√x + x⁻¹) / (3x − 7)

Textbook Question

Use the formal definitions from Exercise 97 to prove the limit statements in Exercises 98–102.

lim x→2⁻ (1 / (x − 2)) = −∞

Textbook Question

Limits as x → ∞ or x → −∞

lim x→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

Textbook Question

Limits as x → ∞ or x → −∞

lim x→∞ √(x² + 1) / (x + 1)

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

Finding Limits

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

h. lim x → ∞ f(x)

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