StudyXY
Advanced Placement /AP Calculus AB: 2.1.4 The Limit Laws, Part I

AP Calculus AB: 2.1.4 The Limit Laws, Part I

Advanced Placement7 CardsCreated 7 days ago

Limits follow the same arithmetic rules as real numbers. You can add, subtract, multiply, or divide limits (if the denominator isn't zero), and constants can be factored out. These properties simplify evaluating complex expressions involving limits.

PrintEmbedImportReport

note

  • Since limits are just numbers, a lot of the properties of real numbers also apply to limits.

  • Taking the limit of a function is an operation, but the resulting limit is just a number. Therefore, it makes sense that limits have a lot of the same properties that numbers do.

  • The limit of a sum of two functions is equal to the sum of the limits.

  • The limit of a difference of two functions is equal to the difference of the limits.

  • The limit of a product of two functions is equal to the product of the limits.

  • The limit of a quotient of two functions is equal to the quotient of the limits, provided that the denominator does not equal zero.

  • The limit of a function multiplied by a constant is equal to the constant multiplied by the limit.
    In addition, the limit of a function raised to a power is equal to the limit raised to that power.

Tap to flip
Space↑↓
←→Navigate
SSpeak
FFocus
1/7

Key Terms

Term
Definition

note

  • Since limits are just numbers, a lot of the properties of real numbers also apply to limits.

  • Taking the limit of a function ...

Hover to peek or log in to view all

Suppose you are told that lim x→1  f(x)=3 and lim x→1  g(x)=−1. What is the value of lim x→1  [f(x)+2g(x)]?

1

Given lim x→cf(x)=2 and lim x→cg(x)=4, evaluate lim x→c[2f(x)−g(x)].

0

Given lim x→ 4 f(x)=2 and lim x→ 4 g(x)=3,evaluate lim x→ 2 [f(x)−g(x)/2f(x)].

The limit cannot be determined from the information given.

Hover to peek or log in to view all

Is the following equation true for all values of x, a, and c ? limx→ a [c⋅f(x)+g(x)]=c[limx→ af(x)+g(x)]

no

Given lim x→2f(x)=3 and lim x→2g(x)=2,evaluate lim x→23f(x)−g(x)/g(x).

7/2

Hover to peek or log in to view all
Log in to view all terms

Related Flashcard Decks

Advanced Placement

AP Calculus AB: 2.1.8 Continuity and Discontinuity

A function is continuous at a point if it’s defined there, its limit exists, and the function’s value equals that limit. Discontinuities occur when one or more of these conditions fail, leading to jump, point (removable), or infinite discontinuities.

13 cards
View Deck
Advanced Placement

AP Calculus AB: 2.1.7 The Squeeze Theorem

The Squeeze Theorem helps find the limit of a function by "squeezing" it between two simpler functions whose limits are known and equal. If the given function is always between those two near a point, then its limit at that point is the same.

13 cards
View Deck
Advanced Placement

AP Calculus AB: 2.1.6 One-Sided Limits

One-sided limits examine a function's behavior from only one direction—left (−) or right (+). A limit exists at a point only if both the left-hand and right-hand limits exist and are equal. If they differ, the overall limit does not exist.

11 cards
View Deck
Advanced Placement

AP Calculus AB: 2.1.3 The Formal Definition of a Limit

The formal definition of a limit uses ε (epsilon) and δ (delta) to precisely describe how a function approaches a value. It states that for every ε > 0, there exists a δ > 0 such that when x is within δ of c (but not equal to c), f(x) is within ε of the limit L.

6 cards
View Deck
Advanced Placement

AP Calculus AB: 2.1.2 Finding Limits Graphically

Finding limits graphically involves observing the y-value a function approaches as x gets close to a specific point from both sides. If the left and right sides approach the same value, the limit exists; if they differ, the limit does not exist.

9 cards
View Deck
Advanced Placement

AP Calculus AB: 2.1.1 Finding Rate of Change over an Interval

The rate of change measures how a function's output varies with changes in input over an interval, often approximated by the average rate of change. Using position functions, it helps analyze motion by calculating how location changes over time.

18 cards
View Deck

Study Tips

  • Press F to enter focus mode for distraction-free studying
  • Review cards regularly to improve retention
  • Try to recall the answer before flipping the card
  • Share this deck with friends to study together
AP Calculus AB: 2.1.4 The Limit Laws, Part I
TermDefinition

note

  • Since limits are just numbers, a lot of the properties of real numbers also apply to limits.

  • Taking the limit of a function is an operation, but the resulting limit is just a number. Therefore, it makes sense that limits have a lot of the same properties that numbers do.

  • The limit of a sum of two functions is equal to the sum of the limits.

  • The limit of a difference of two functions is equal to the difference of the limits.

  • The limit of a product of two functions is equal to the product of the limits.

  • The limit of a quotient of two functions is equal to the quotient of the limits, provided that the denominator does not equal zero.

  • The limit of a function multiplied by a constant is equal to the constant multiplied by the limit.
    In addition, the limit of a function raised to a power is equal to the limit raised to that power.

Suppose you are told that lim x→1  f(x)=3 and lim x→1  g(x)=−1. What is the value of lim x→1  [f(x)+2g(x)]?

1

Given lim x→cf(x)=2 and lim x→cg(x)=4, evaluate lim x→c[2f(x)−g(x)].

0

Given lim x→ 4 f(x)=2 and lim x→ 4 g(x)=3,evaluate lim x→ 2 [f(x)−g(x)/2f(x)].

The limit cannot be determined from the information given.

Is the following equation true for all values of x, a, and c ? limx→ a [c⋅f(x)+g(x)]=c[limx→ af(x)+g(x)]

no

Given lim x→2f(x)=3 and lim x→2g(x)=2,evaluate lim x→23f(x)−g(x)/g(x).

7/2

Determine the limit (if it exists):
lim x→0 sinx/2x
Hint: lim x→0 sinx/x=1

1/2