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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2:15 minutes

Problem 2.6.23

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))

Verified step by step guidance

Identify the highest power of x in the denominator. In this case, the highest power is x².

Divide both the numerator and the denominator by x². This simplifies the expression to √((8 - 3/x²) / (2 + 1/x)).

As x approaches ∞, the terms 3/x² and 1/x approach 0 because they contain x in the denominator, which becomes very large.

Substitute these limits into the simplified expression: √((8 - 0) / (2 + 0)) = √(8/2).

Simplify the expression inside the square root: √(8/2) = √4.

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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 often requires simplifying the function to identify dominant terms, which dictate the function's end behavior. Understanding limits at infinity is crucial for analyzing asymptotic behavior and determining horizontal asymptotes of rational functions.

Rational Functions

A rational function is a ratio of two polynomials. To find limits involving rational functions, especially as x approaches infinity, it's essential to compare the degrees of the numerator and denominator. The highest degree terms often dominate the behavior of the function, simplifying the process of finding limits at infinity.

Simplifying Expressions

Simplifying expressions, particularly those involving square roots and polynomials, is key to solving limit problems. By dividing the numerator and denominator by the highest power of x in the denominator, we can reduce the expression to a simpler form. This technique helps in identifying the leading terms and determining the limit as x approaches infinity.

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Related Practice

Textbook Question

Finding Limits

In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)

g(x) = 1/(2 + (1/x))

Textbook Question

Finding Limits

In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)

h(x) = (−5 + (7/x))/(3 – (1/x²))

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 → ⁻∞ ((1 − x³) / (x² + 7x))⁵

Textbook Question

Limits as x → ∞ or x → −∞

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)

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