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

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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2:22 minutes

Problem 2.42

Textbook Question

Limits and Infinity

Find the limits in Exercises 37–46.

x⁴ + x³
lim -----------------
x→∞ 12x³ + 128

Verified step by step guidance

Identify the highest power of x in both the numerator and the denominator. In this case, the highest power in the numerator is x⁴ and in the denominator is x³.

Divide every term in the numerator and the denominator by x³, the highest power of x in the denominator.

Rewrite the expression: \( \lim_{{x \to \infty}} \frac{{x^4/x^3 + x^3/x^3}}{{12x^3/x^3 + 128/x^3}} \). This simplifies to \( \lim_{{x \to \infty}} \frac{{x + 1}}{{12 + \frac{128}{x^3}}} \).

As x approaches infinity, the term \( \frac{128}{x^3} \) approaches 0 because the denominator grows much faster than the numerator.

Evaluate the limit: \( \lim_{{x \to \infty}} \frac{{x + 1}}{{12}} \). As x approaches infinity, the dominant term is x, so the limit simplifies to \( \frac{x}{12} \), which approaches infinity.

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Key Concepts

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the behavior of the function as x approaches infinity, which helps us understand the end behavior of rational functions.

Rational Functions

A rational function is a ratio of two polynomials. In the given limit problem, the numerator is a polynomial of degree 4, and the denominator is a polynomial of degree 3. The degrees of the polynomials play a crucial role in determining the limit as x approaches infinity.

Dominant Terms

In limits involving polynomials, the dominant term is the term with the highest degree, as it has the most significant impact on the function's value as x approaches infinity. For the given expression, the dominant term in the numerator is x⁴, and in the denominator, it is 12x³, which will dictate the limit's value.

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Textbook Question

{Use of Tech} Computing limits with angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.

b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.