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AP Calculus AB: 2.2.4 An Overview of Limits

Mathematics11 CardsCreated 3 months ago

This flashcard set provides a broad overview of limits, covering fundamental concepts like indeterminate forms, algebraic simplification, and graphical interpretation. It also explores continuity, types of discontinuities, and special trigonometric limits through both symbolic and applied examples.

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An Overview of Limits

  • The limit is the range value that a function approaches as you get closer to a particular domain value.

  • An indeterminate form is a mathematically meaningless expression.

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Key Terms

Term
Definition

An Overview of Limits

  • The limit is the range value that a function approaches as you get closer to a particular domain value.

  • An indeterminate for...

note

  • This limit involves an unusual variable.

  • Remember to use direct substitution as a first step in evaluating limits. In this c...

LetG(x)= x^2−4/x+2, x≠−2
k, x=−2
Find the value of k so that lim x→−2 G(x)=G(−2).

-4

Classify all of the discontinuities of the function h(x)=f(g(x)) given f(x)=1/x−3 and g(x)=x^2+2.

x = −1 and x = 1; infinite discontinuities

Given that lim x→0(sinx)^2/x=0, find the limit.

lim x→0 1−cosx/x

0

Evaluate the limit limCOW→3

[4(COW)−12 / (COW)^2+(COW)−12].

4/7

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AP Calculus AB: 2.2.4 An Overview of Limits
TermDefinition

An Overview of Limits

  • The limit is the range value that a function approaches as you get closer to a particular domain value.

  • An indeterminate form is a mathematically meaningless expression.

note

  • This limit involves an unusual variable.

  • Remember to use direct substitution as a first step in evaluating limits. In this case, direct substitution produces the familiar indeterminate formof 0/0.

  • Proceed by factoring the numerator, which is a difference of two squares.

  • Use cancellation to simplify the limit expression and then apply direct substitution to arrive at the result.

  • The existence of limits can be demonstrated graphically. On the far left, the graph shows that near x= 7 the function is approaching the same value from both the left and the right. The limit exists and equals that value, even though the function takes on a different value at x= 7.

  • On the near left, the graph approaches different values on either side of x= 5. Since the two one-sided limits have different values, the limit of the function does not exist.

  • Here is an example of a function that is approaching very large values from the one side and very small values from the other. The limit for such a function does not exist.

LetG(x)= x^2−4/x+2, x≠−2
k, x=−2
Find the value of k so that lim x→−2 G(x)=G(−2).

-4

Classify all of the discontinuities of the function h(x)=f(g(x)) given f(x)=1/x−3 and g(x)=x^2+2.

x = −1 and x = 1; infinite discontinuities

Given that lim x→0(sinx)^2/x=0, find the limit.

lim x→0 1−cosx/x

0

Evaluate the limit limCOW→3

[4(COW)−12 / (COW)^2+(COW)−12].

4/7

Does f (x) have a limit at x = −3?

No, the limit doesn’t exist.

Evaluate the limit

limΔx→0 4(Δx+2)^2+5Δx−3/6Δx+1

13

If f(x)=4x2−4xx+1,

evaluate the limit lim x→−1 f(x).

The limit does not exist.

Given the limit lim x→2(2x+2)=6, what is the largest value of δ such that ε

.005

Given the limit lim x→1 (4x+3)=7,what is the largest value of δ such that ε≤.01?

.0025