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

Previous problemNext problem

2:02 minutes

Problem 2.6.37

Textbook Question

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0⁺ 1 / 3x

Verified step by step guidance

Step 1: Understand the concept of infinite limits. An infinite limit occurs when the value of a function increases or decreases without bound as the input approaches a certain point.

Step 2: Analyze the given function, which is \( \frac{1}{3x} \). As \( x \) approaches 0 from the right (denoted by \( x \to 0^+ \)), the value of \( 3x \) becomes a very small positive number.

Step 3: Consider the behavior of the function \( \frac{1}{3x} \) as \( x \to 0^+ \). Since \( 3x \) is a small positive number, \( \frac{1}{3x} \) becomes a large positive number.

Step 4: Conclude that as \( x \to 0^+ \), \( \frac{1}{3x} \) tends towards positive infinity. Therefore, the limit is \( \infty \).

Step 5: Write the final expression for the limit: \( \lim_{x \to 0^+} \frac{1}{3x} = \infty \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit of the function as x approaches 0 from the positive side.

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. This can happen when the function approaches a vertical asymptote. In the given limit, we need to determine whether the function approaches positive or negative infinity as x approaches 0 from the right.