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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Multiple Choice

Which of the following explains why a function $f (x)$ is discontinuous at $x = a$ ?

The limit of

f (x)

x

approaches

a

does not exist.

The limit of

f (x)

x

approaches

a

exists, but is not equal to

f (a)

f (a)

is not defined.

f (a)

is defined and the limit of

f (x)

x

approaches

a

equals

f (a)

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Verified step by step guidance

Step 1: Recall the definition of continuity at a point x = a. A function f(x) is continuous at x = a if the following three conditions are satisfied: (1) f(a) is defined, (2) the limit of f(x) as x approaches a exists, and (3) the limit of f(x) as x approaches a is equal to f(a).

Step 2: Analyze the first condition: f(a) must be defined. If f(a) is not defined, the function is automatically discontinuous at x = a.

Step 3: Analyze the second condition: The limit of f(x) as x approaches a must exist. If the left-hand limit (as x approaches a from the left) and the right-hand limit (as x approaches a from the right) are not equal, the limit does not exist, and the function is discontinuous at x = a.

Step 4: Analyze the third condition: The limit of f(x) as x approaches a must equal f(a). If the limit exists but is not equal to f(a), the function is discontinuous at x = a.

Step 5: Summarize the reasons for discontinuity: A function f(x) is discontinuous at x = a if (1) f(a) is not defined, (2) the limit of f(x) as x approaches a does not exist, or (3) the limit exists but is not equal to f(a).