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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2:07 minutes

Problem 59a

Textbook Question

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim (2 − 3 / t¹/³) as

a. t → 0⁺

Verified step by step guidance

Identify the expression for which you need to find the limit: \( 2 - \frac{3}{t^{1/3}} \).

Recognize that as \( t \to 0^+ \), \( t^{1/3} \to 0^+ \) as well, since the cube root of a positive number approaching zero also approaches zero.

Consider the behavior of the term \( \frac{3}{t^{1/3}} \). As \( t^{1/3} \to 0^+ \), \( \frac{3}{t^{1/3}} \to +\infty \) because dividing by a very small positive number results in a very large positive number.

Analyze the entire expression \( 2 - \frac{3}{t^{1/3}} \). Since \( \frac{3}{t^{1/3}} \to +\infty \), the expression \( 2 - \frac{3}{t^{1/3}} \) will tend towards \(-\infty\).

Conclude that the limit of the expression as \( t \to 0^+ \) is \(-\infty\).

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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 concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near specific points, including points of discontinuity or infinity. In this case, we are interested in the limit of the function as t approaches 0 from the positive side (0⁺).

Cube Root Function

The cube root function, denoted as t¹/³, is a continuous function that returns the number which, when cubed, gives the input value. As t approaches 0, the cube root of t also approaches 0. Understanding how this function behaves near 0 is essential for evaluating the limit in the given problem.