Limits Flashcards
What is the limit as x approaches 2 for the function -x + 6?
4
This is calculated by substituting x with 2 in the function.
When substituting values in a limit function, what happens if the equation is a fraction?
It is not as simple as plugging in values directly because you would get 0/0
You may need to analyze the behavior from both sides of the limit.
What is the limit as x approaches 2 for the fraction (x-4)/(x-2)?
4
Approaching from the left and right sides yields values that converge to 4.
Fill in the blank: As x approaches 2, the limit of (2.1-4)/(2.1-2) approaches _______.
4.1
Fill in the blank: As x approaches 2, the limit of (2.01-4)/(2.01-2) approaches _______.
4.01
Fill in the blank: As x approaches 2, the limit of (1.93-4)/(1.93-2) approaches _______.
3.9
What does it mean when a limit converges into a certain value?
It approaches a specific value as the input gets closer to a point
In this case, the limit approaches 4 as x approaches 2.
What is one method to find the limit of a function if direct substitution is not possible?
Factorising the function
This can help simplify the expression to evaluate the limit.
What is the limit as x approaches 2 of (x³ - 4)/(x - 2)?
2 + 2 = 4
This limit can be evaluated by factorizing the numerator.
In the expression (3 - x)(x - 3), what is the relationship between (3 - x) and (x - 3)?
(3 - x) = -1(x - 3)
This shows that they are negatives of each other.
True or False: You can directly substitute x = 3 into (x - 3)/(x - 3) without simplification.
False
The expression is indeterminate at x = 3 and requires simplification.
Fill in the blank: The limit as x approaches 3 of (3 - x)/(x - 3) equals _______.
-1
What is the first step to solve a problem involving a limit with a square root?
Multiply the top and the conjugate of the square root
What is the limit notation for approaching from the left?
lim x→a-
Indicates the limit as x approaches a from the left side.
What is the limit notation for approaching from the right?
lim x→a+
Indicates the limit as x approaches a from the right side.
What does it mean if lim x→a- and lim x→a+ are equal?
lim x→a exists
The limit exists if both one-sided limits match.
What does it mean if lim x→a- and lim x→a+ are different?
Limit does not exist (DNE)
If the left-hand limit and right-hand limit do not match, the limit is undefined.
How is f(a) determined when discussing limits?
Closed dot’s y-value
The value of the function at x=a is determined by the closed dot on the graph.
What happens if the x-value is near a vertical asymptote?
Check limits before and after
The behavior of the function around vertical asymptotes can affect the limit.
What is the result of lim x→1 if it approaches from the left and right yield different values?
DNE
This indicates the limit does not exist when the left and right limits are unequal.
Fill in the blank: To find the limit as x approaches a, if the previous values match, write down that value. If they are different, the value does not exist (____).
DNE
True or False: The limit can exist even if the function is not defined at that point.
True
Limits can exist independently of the function’s value at that point.
What is a vertical asymptote?
A vertical asymptote occurs when the function approaches infinity or negative infinity as the x-value approaches a certain point
It indicates that the function is undefined at that point.
What does lim f(x) as x approaches 1 from the left equal if the vertical asymptote is going down?
-∞
This indicates that the function decreases without bound as it approaches the asymptote.
What does lim f(x) as x approaches 1 from the right equal if the vertical asymptote is going up?
+∞
This indicates that the function increases without bound as it approaches the asymptote.
What is a limit defined as continuous?
- f(a) is defined
- lim f(x) exists as x approaches a
- Lim f(x) = f(a) as x approaches a
These conditions ensure the function behaves consistently at the point of interest.
What is the value of lim x→0 of x?
+∞
This indicates that as x approaches 0 from the right, the function grows without bound.
What is the value of lim x→0 of x from the left?
-∞
This shows that as x approaches 0 from the left, the function decreases without bound.
What does lim x→+∞ of 1+x equal?
+∞
This indicates that as x grows larger, the value of the function continues to increase without bound.
What does it mean when a function is bottom heavy?
It equals O
Bottom heavy functions approach zero as they grow.
What does it mean when a function is top heavy?
It equals +∞
Top heavy functions diverge to infinity.
What theorem is used to find the limit of a function?
Squeeze theorem
The Squeeze theorem helps determine limits by comparison.
What is the form of the Squeeze theorem?
If lim h(x) < lim f(x) < lim g(x), then lim f(x) = L
This applies as x approaches a specific value.
In the Squeeze theorem, if lim h(x) = L and lim g(x) = L, what can be concluded about lim f(x)?
lim f(x) = L
This establishes that f(x) is squeezed to L.
What is the limit of 8 - x³ as x approaches 0?
8
Evaluating the limit gives 8 - 0³ = 8.
What is the limit of 8 + x³ as x approaches 0?
8
Evaluating the limit gives 8 + 0³ = 8.
What conclusion can be drawn from lim 8 - x³ < lim f(x) < lim 8 + x³?
lim f(x) = 8
This demonstrates that f(x) converges to 8 as x approaches 0.
