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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2:45 minutes

Problem 2.6.27

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)

Verified step by step guidance

Identify the highest power of x in the denominator, which is x in this case.

Divide every term in the numerator and the denominator by x, the highest power of x in the denominator.

Rewrite the expression: (2√x/x + x⁻¹/x) / (3x/x - 7/x).

Simplify each term: (2/√x + 1/x²) / (3 - 7/x).

Evaluate the limit as x approaches infinity: As x → ∞, 2/√x → 0, 1/x² → 0, and 7/x → 0, so the limit simplifies to 0/3.

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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 refer to the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, particularly for rational functions where the degree of the numerator and denominator can determine the limit. Analyzing limits at infinity helps in identifying horizontal asymptotes and the end behavior of functions.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The degree of the polynomials in the numerator and denominator plays a significant role in determining the limit as x approaches infinity. Understanding the structure of rational functions allows for the simplification of limits by dividing through by the highest power of x, which reveals the dominant terms that dictate the limit's value.

Dominant Terms

In the context of limits, dominant terms are the terms in a polynomial that have the highest degree and thus have the most significant impact on the function's behavior as x approaches infinity. When evaluating limits, identifying and focusing on these terms allows for simplification of the expression, making it easier to determine the limit. This concept is essential for effectively applying the technique of dividing by the highest power of x.

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Related Practice

Textbook Question

Limits of Rational Functions

In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.

f(x) = (2x + 3)/(5x + 7)

Textbook Question

Limits of Rational Functions

In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.

f(x) = (x + 1)/(x² + 3)

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 → ∞ √((8x² − 3) / (2x² + x))

Textbook Question

Limits as x → ∞ or x → −∞

lim x → ⁻∞ ((1 − x³) / (x² + 7x))⁵

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 → −∞