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Mathematics /AP Calculus AB: 2.1.4 The Limit Laws, Part I

AP Calculus AB: 2.1.4 The Limit Laws, Part I

Mathematics7 CardsCreated 7 months 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.

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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.

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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 ...

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

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AP Calculus AB: 2.1.4 The Limit Laws, Part I

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 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