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AP Calculus AB > Unit 1: Limits and Continuity > Flashcards

Unit 1: Limits and Continuity Flashcards

1
Q

What is the definition of a limit?

A

A limit is the value that a function approaches as the input approaches a certain point.

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2
Q

True or False: The limit of a function can be different from the function’s value at that point.

A

True

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3
Q

Fill in the blank: The notation for the limit of f(x) as x approaches a is ______.

A

lim(x→a) f(x)

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4
Q

What is the Squeeze Theorem used for?

A

It is used to find limits of functions that are squeezed between two other functions.

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5
Q

What does it mean for a limit to be infinite?

A

It means that the function increases or decreases without bound as it approaches a certain point.

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6
Q

What is the formal definition of continuity at a point?

A

A function is continuous at a point if the limit as x approaches the point equals the function’s value at that point.

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7
Q

Multiple choice: Which of the following is a requirement for a function to be continuous at x = c? A) f(c) exists B) lim(x→c) f(x) exists C) lim(x→c) f(x) = f(c) D) All of the above

A

D) All of the above

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8
Q

What is an asymptote?

A

An asymptote is a line that a graph approaches but never touches.

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9
Q

True or False: Vertical asymptotes occur where a function approaches infinity.

A

True

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10
Q

What is the difference between a removable and a non-removable discontinuity?

A

A removable discontinuity can be ‘fixed’ by redefining the function at that point, while a non-removable discontinuity cannot be fixed.

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11
Q

What is the Intermediate Value Theorem?

A

The IVT states that if f is continuous on the interval [a,b] and N is a value between f(a) and f(b), then there exists at least one c in (a,b) such that f(c)=N.

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12
Q

Multiple choice: If a function has a limit of 0 as x approaches 2, what can we conclude? A) f(2) = 0 B) f(x) approaches 0 as x approaches 2 C) f(2) does not exist D) All of the above

A

B) f(x) approaches 0 as x approaches 2

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13
Q

What does it mean for a limit to exist?

A

A limit exists if the left-hand limit and the right-hand limit as 𝑥 approaches a point 𝑐 are equal.

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14
Q

Define a one-sided limit.

A

A one-sided limit is the value a function approaches as the input approaches a specific point from either the left or right.

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15
Q

What are the three types of discontinuities?

A
  1. Removable Discontinuity (hole in the graph)
  2. Jump Discontinuity (sudden jump in values)
  3. Infinite Discontinuity (vertical asymptote)
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16
Q

Squeeze Theorem

A

f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them

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17
Q

How do you evaluate limits at infinity?

A

To evaluate limits at infinity, analyze the end behavior of the function as x→∞ or x→−∞ by comparing the degrees of the terms in the numerator and denominator (for rational functions).

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18
Q

What is the difference between horizontal and vertical asymptotes?

A

Horizontal asymptotes occur as x→±∞ and indicate the end behavior of a function.

Vertical asymptotes occur at values of 𝑥 where the function approaches infinity or negative infinity, typically where the denominator is zero in a rational function.

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19
Q

State the Constant Multiple Law for limits.

A

If lim x→c f(x)=L and 𝑘 is a constant, then lim x→c k⋅f(x)=k⋅L

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20
Q

What is the Sum Law for limits?

A

If lim x→c f(x) = L and lim x→c g(x) = M, then lim (f(x) + g(x)) = L + M

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21
Q

What is the Difference Law for limits?

A

If lim x→c f(x) = L and lim x→c g(x) = M, then lim x→c (f(x) − g(x)) = L − M

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22
Q

What is the Constant Multiple Law for limits?

A

If lim x→c f(x) = L and k is a constant, then lim x→c (k ⋅ f(x)) = k ⋅ L

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23
Q

What is the Product Law for limits?

A

If lim x→c f(x) = L and lim x→c g(x) = M, then lim x→c (f(x) ⋅ g(x)) = L ⋅ M

24
Q

What is the Quotient Law for limits?

A

If lim x→c f(x)=L and lim x→c g(x)=M, and M ≠0, then lim x→c f(x) / g(x) = L / M

25
Q

What is the Power Law for limits?

A

If lim x→c f(x) = L and a is a positive integer, then lim x→c
​(f(x))ª =Lª

26
Q

What is the Root Law for limits?

A

If lim x→c f(x) = L and a is a positive integer, then lim x→a ª√f(x) = ª√L, provided L≥0 when a is even

27
Q

lim x-> 0 sin ax/ax = 1

A

One of two special limits

28
Q

lim x-> 0 (1 - cos(ax)) / ax

A

One of two special limits

29
Q

Types or limit discontinuities

A
  1. Jump Discontinuity
  2. Infinite Discontinuity
  3. Removable Discontinuity
  4. Essential Discontinuity
30
Q

How to determine a Jump Discontinuity

A

the lim x->a - f(x) ≠ the lim a->a + f(x)

31
Q

How to determine a Infinite Discontinuity

A

Check if lim x→a −f(x) or lim x→a + f(x) tends to ∞
or −∞, if yes then there is an infinite discontinuity at x=a.

32
Q

How to determine a Removable Discontinuity

A
  1. Check if the limit lim x→a f(x) exists
  2. Check if the function is either not defined at x=a, or if it is defined but f(a) ≠ lim x→a f(x)
33
Q

How to determine a Essential Discontinuity

A

If the function oscillates wildly as x→a such as trigonometric functions

34
Q

Rationalizing the numerator

A

Multiple the numerator containing the root (ex. √(x+7) + 1) by its conjugate (√(x+7) - 1) to the num. and denominator

35
Q

When is rationalizing the numerator necessary

A

help simplify expressions when calculating limits, especially when dealing with indeterminate forms like 0/0

36
Q

What does 0/0 mean

A

Indeterminate form (M0RE / W0RK) like
1. Simplify the Expression (Algebraic Manipulation)
2. Factorization
3. Rationalization
4. Special limits
5. L’Hopital’s Rule

37
Q

limit of a function as
x approaches a number involves…

A

determining the value that the function approaches as x gets arbitrarily close to a particular point

38
Q

Steps to solve for a limit of a function as x approaches a number

A
  1. Direct Substitution (if possible)
  2. If = 0/0 then use factorization, simplification, Rationalization
  3. Check for One-Sided Limits (if needed)
39
Q

limit as x approaches infinity involves…

A

determining the behavior of the function as x becomes arbitrarily large

40
Q

Steps to solve for a limit of a function as x approaches infinity

A

Determine the highest power of f(x) = p(x) / q(x)
- if p(x) < q(x), then y = 0
- If p(x) = q(x), the horizontal asymptote is y = leading coefficient of num / leading coefficient of den
- if p(x) > q(x), then the function may approach infinity or negative infinity (no horizontal asymptote)

41
Q

How to calucalate the limit of piecewise functions at a particular point

A

look at the left-hand limit and right-hand limit. If these two one-sided limits match, then the limit exists. If they do not match, the two-sided limit does not exist at that point.

42
Q

Steps to find piecewise limits

A
  1. Identify the point of interest where you want to find the limit
  2. Check lim x→a −f(x)
  3. Check lim x→a +f(x)
  4. Compare the left-hand and right-hand limits
43
Q

A piecewise function is continuous at a point
c if…

A

The limit as x→c exists ( the left-hand limit and right-hand limit are equal).
2. The function value at x=c exists (f(c) is defined).
3. The limit equals the function value (lim x→c f(x)=f(c))

44
Q

Steps to Make a Piecewise Function Continuous

A
  1. Find lim x→a −f(x)
  2. Find lim x→a +f(x)
  3. Set the function value x=c to be the same as the common limit
  4. If the function has a gap or discontinuity, adjust the value of f(c) so that: lim x→c f(x)=f(c)
45
Q

Steps to find Find lim x→a −f(x)

A
  1. Identify the function and the point of interest 𝑎
  2. determine how the function behaves as x approaches
    𝑎 from values smaller than 𝑎 (consider values like 𝑎−0.1,𝑎−0.01,…)
  3. Check for continuity or discontinuity
46
Q

Steps to find Find lim x→a +f(x)

A
  1. Identify the function and the point of interest 𝑎
  2. determine how the function behaves as x approaches
    𝑎 from values bigger than 𝑎 (consider values like 𝑎+0.1,𝑎+0.01,…)
  3. Check for continuity or discontinuity
47
Q

Formula and steps for the Limit of a Composition

A

Lim x→a f(g(x)) = f(lim x→a g(x))
1. Find the limit of the inner function
g(x) as x→a
2. Substitue the value
lim x→a g(x) into the function f(x)

48
Q

limit of the composition of functions refers to…

A

first find the limit of g(x) as x→a, and then evaluate the outer function f(x) at that limit.

49
Q

limit of the composition of functions is valid if…

A

the inner function g(x) approaches a value that is within the domain of the outer function f(x). If
lim x→a g(x) is outside the domain of f(x), the limit does not exist.

50
Q

Average Rate Or Change is…

A

Slope between two points on a curve

51
Q

AROC Formula on [a,b]

A

[f(b) - f(a)] / (b - a)

52
Q

Instantaneous Rate of Change (IROC) is…

A

the slope of the line tangent to the curve at a single point (x=a)

53
Q

IROC Formula at x = a

A

m = lim h→0 (f(a + h) - f(a)) / h

54
Q

What are horizontal asymptotes in limits?

A

a horizontal asymptote indicates the end behavior of a function—what the function “settles” toward as x moves farther away from the origin (to infinity or negative infinity).

55
Q

A function f(x) has a horizontal asymptote at
y=L if…

A

as x approaches ∞ or −∞, the function values approach the constant L

56
Q

Steps to find horizontal asymptotes for Rational Functions: If f(x) = q(x) / p(x) and both are polynomial

A
  • if p(x) < q(x), the horizontal asymptote is y = 0
  • If p(x) = q(x), the horizontal asymptote is y = leading coefficient of num / leading coefficient of den
  • if p(x) > q(x), then there i no horizontal asymptote (the function may approach infinity or negative infinity)
    (Same as finding limits with infinities)

AP Calculus AB (3 decks)

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