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

Continuity

Calculus

1. Limits and Continuity

Continuity

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2:41 minutes

Problem 2.5.18

Textbook Question

At what points are the functions in Exercises 13–30 continuous?
y = 1/(|x| + 1) − x²/2

Verified step by step guidance

Step 1: Understand the concept of continuity. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.

Step 2: Break down the function y = 1/(|x| + 1) − x²/2 into its components. The function consists of two parts: 1/(|x| + 1) and −x²/2.

Step 3: Analyze the continuity of the first part, 1/(|x| + 1). The expression |x| represents the absolute value of x, which is continuous everywhere. The denominator |x| + 1 is never zero, so 1/(|x| + 1) is continuous for all x.

Step 4: Analyze the continuity of the second part, −x²/2. The function x² is a polynomial, and polynomials are continuous everywhere. Therefore, −x²/2 is continuous for all x.

Step 5: Combine the results from steps 3 and 4. Since both components of the function are continuous for all x, the entire function y = 1/(|x| + 1) − x²/2 is continuous for all x.

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Key Concepts

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

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph of the function at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. Understanding how these functions behave at the boundaries of their defined pieces is crucial for determining continuity. In the given function, the absolute value and polynomial components may create different behaviors depending on the value of x.

Limits

Limits are fundamental in calculus for analyzing the behavior of functions as they approach specific points. To determine continuity, one must evaluate the limit of the function as x approaches a point from both the left and right. If these limits exist and are equal to the function's value at that point, the function is continuous there.