Limit Notation

Show your work for each problem.

1.
How do you read ?
1. a) equals in
2. b)The function equals
3. c)The limit of as approaches
4. d)The value of
Answer: The limit of as approaches
The notation reads "the limit of as approaches ." The arrow means "approaches," not "equals."
2.
What does the arrow () mean in limit notation?
1. a)Transforms into
2. b)Is greater than
3. c)Approaches (gets arbitrarily close to)
4. d)Equals
Answer: Approaches (gets arbitrarily close to)
The arrow means "approaches" or "gets arbitrarily close to." When we write , we mean gets closer and closer to , but never actually equals .
3.
In the expression , what value is approaching?
Answer: 7
The subscript tells us that is approaching . This is read as "the limit as approaches ."
4.
What does mean?
1. a)The limit as approaches from the right
2. b) equals plus something
3. c)The limit as approaches from the left
4. d)The limit as approaches positive
Answer: The limit as approaches from the right
The notation means "from the right" or "from values greater than ." So means approaches through values like
5.
If , what is ?
Answer: 10
As approaches , the expression approaches . For continuous functions like polynomials, the limit equals the function value at that point.
6.
What does represent?
1. a)The left-hand limit at
2. b)The negative limit at
3. c)The limit of
4. d)The right-hand limit at
Answer: The left-hand limit at
The notation means approaching from the left (from values less than ), like . This is called the left-hand limit.
7.
Write the limit notation for: "The limit of as approaches equals ."
Answer: lim(x→3)(x²+1)=10
1. What symbol starts a limit expression? lim
2. What value does approach? 3
3. What function are we taking the limit of? x^2+1
4. What does the limit equal? 10
8.
What does mean?
1. a)When ,
2. b)The function has a maximum value of
3. c)As grows without bound, approaches
4. d) grows to infinity at
Answer: As grows without bound, approaches
This notation says that as becomes arbitrarily large (grows without bound), the function values get closer and closer to . The line is a horizontal asymptote.
9.
In , from which direction does approach ?
Answer: right
The notation means approaching from the right (positive side). So takes values like which are all positive and get closer to from the right on the number line.
10.
Translate to limit notation: "As approaches from the left, approaches ."
Answer: lim(x→2⁻)f(x)=-∞
1. What is the limit operator symbol? lim
2. What value does approach? 2
3. What superscript indicates 'from the left'? -
4. What does approach? -∞
11.
What does it mean if ?
1. a)The function is continuous at
2. b)The function equals at
3. c)The two-sided limit does not exist
4. d)The one-sided limits are both infinite
Answer: The two-sided limit does not exist
For a two-sided limit to exist, the left-hand and right-hand limits must be equal. If they're different, we say the limit does not exist (DNE) at that point.
12.
Interpret the statement:
Answer: As x decreases without bound, the function approaches 3
1. What does mean? x decreases without bound
2. What is the function we're analyzing? 3x/(x+1)
3. What value does the function approach? 3
4. What type of asymptote does this represent? horizontal
13.
Which statement correctly interprets ?
1. a)The sine of divided by equals
2. b)When , the function equals
3. c)
4. d)As approaches , the ratio approaches
Answer: As approaches , the ratio approaches
The function is undefined at (division by zero!), but as gets closer to (from either side), the values approach . This is a fundamental limit in calculus.
14.
If and , what is ?
Answer: 4
Since both one-sided limits equal , the two-sided limit exists and equals . This is the definition: if and only if .
15.
Write the complete limit notation for: "As time increases without bound, the population approaches its carrying capacity of ."
Answer: lim(t→∞)P(t)=1000
1. What variable represents time in the problem? t
2. What does 'increases without bound' translate to in limit notation? ∞
3. What function describes the population? P(t)
4. What value does the population approach? 1000
16.
What is wrong with writing just ""?
1. a)It doesn't specify what value approaches
2. b)The function name should be in parentheses
3. c)The equal sign is not needed
4. d)The limit value should be written first
Answer: It doesn't specify what value approaches
The subscript showing what value approaches is essential. Without it, we don't know if we're finding the limit as , , , etc. The correct form requires with a specific value for .