Introduction to Limits

Show your work for each problem.

1.
What does mean?
1. a) approaches 3 as approaches 5
2. b)
3. c) equals 5 when equals 3
4. d)
Answer: approaches 3 as approaches 5
The notation means that as gets closer and closer to 5, the value of gets closer and closer to 3. This is different from saying .
2.
Find
Answer: 10
For continuous functions like , the limit equals the function value: .
3.
Find
Answer: 5
Direct substitution works: .
4.
What form do you get when you try by direct substitution?
1. a)
2. b)
3. c)
4. d)
Answer:
Numerator: . Denominator: . This gives , an indeterminate form that requires factoring.
5.
Find
Answer: 6
1. Factor the numerator . What do you get? (x+3)(x-3)
2. After factoring, cancel the common factor. What simplified expression remains? x+3
3. Now substitute into the simplified expression. 6
6.
Find
Answer: 2
. So .
7.
Which statement is true about if ?
1. a)The limit equals 0
2. b)The limit equals 4, even though is undefined
3. c)The limit equals
4. d)The limit does not exist
Answer: The limit equals 4, even though is undefined
for . The limit is , but is undefined (division by zero). This shows limits can exist at points where the function is undefined!
8.
Find
Answer: 1
1. Factor the numerator . What are the factors? (x+1)(x+2)
2. After factoring and canceling , what expression remains? x+2
3. Substitute into the simplified expression. 1
9.
What is ?
1. a)Does not exist
2. b)
3. c)
4. d)
Answer:
As approaches 0 from the right (positive side), becomes larger and larger: , , etc. So .
10.
Find
Answer: 2.5
. As , and , so the limit is .
11.
Find
Answer: 0.5
1. What is the highest power of in the expression? x^2
2. After dividing all terms by , what is the new numerator? 2 - 3/x + 1/x^2
3. As , what do , , and all approach? 0
4. So the limit simplifies to 0.5
12.
For , what can we say about ?
1. a)The limit equals 1
2. b)The limit equals 0
3. c)DNE because and
4. d)The limit equals -1
Answer: DNE because and
From the right: . From the left: . Since one-sided limits differ, the two-sided limit does not exist.
13.
Find
Answer: infinity
When the numerator's degree (3) is higher than the denominator's degree (2), the limit is . The numerator grows faster than the denominator.
14.
Find
Answer: 0.25
1. Note that can be factored as . What is this technique called? difference of squares
2. Rewrite the expression using this factorization. What simplifies? sqrt(x) - 2
3. The simplified expression is . Substitute . 0.25
15.
True or False: If , then .
1. a)False
2. b)True
Answer: False
FALSE. The limit can exist even when is undefined or has a different value. For example, , but is undefined.
16.
Find
Answer: 5/3
1. Factor the numerator . (x-5)(x+5)
2. Factor the denominator . (x-5)(x+1)
3. After canceling , what is the simplified expression? (x+5)/(x+1)
4. Substitute into the simplified expression. 5/3
17.
If and , find
Answer: 15
By the limit product rule: .
18.
Find
Answer: 5
For polynomials, direct substitution always works: .