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Limits Evaluating Limits Algebraically – Direct Substitution
Worksheet
4
Evaluating Limits Algebraically – Direct Substitution If the limit exists, evaluate. If the limit doesn’t exist, write DNE. If you cannot determine the answer using direct substitution, classify it as an indeterminate.
1. lim$→&
𝑥 − 3𝑥 + 2
2. lim$→&
𝑥 + 2𝑥 − 2
3. lim$→&
1/𝑥 − 1/2𝑥 − 2
4. lim$→&
𝑥 − 2𝑥 + 2
5. lim$→&
2$ − 1 6. lim$→.
𝑥/ − 2𝑥 − 8
7. lim$→12
13
&$
8.
lim$→3
2𝑥 + 1 − 3𝑥& − 16
9. lim
$→12/&
𝑥 − 22𝑥 + 1 =
10. lim$→3
tan𝜋𝑥 11. lim
$→1:𝑥 sec
𝜋𝑥 12. lim
$→>
sin 𝑥𝑥
13. lim$→?/&
tan(3𝑥) 14. lim
$→2/&
cos12 𝑥𝑥
15. lim$→?
sin12 𝑥
16. lim$→2>>
log 𝑥 17. lim$→:
log&(𝑥& − 1)𝑥 − 3
18. lim$→2
log 𝑥𝑥& − 1
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Limits Evaluating Limits Algebraically – Factoring
Worksheet
5
Evaluating Limits Algebraically – Factoring Evaluate:
1. lim$→>
10𝑥20𝑥& + 15𝑥
2. lim$→2/:
6𝑥& + 𝑥 − 11 − 3𝑥
3. lim$→G
𝑥& + 𝑥 − 56𝑥& − 13𝑥 + 42
4.
lim$→1&
−3𝑥 + 𝑥& − 10−𝑥& − 10 − 7𝑥
5. lim
$→1&/:
36𝑥: − 16𝑥3𝑥& − 𝑥 − 2
6.
lim$→1&
𝑥: + 2𝑥& − 36𝑥 − 72𝑥& − 4𝑥 − 12
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Limits Evaluating Limits Algebraically – Common Denominators
Worksheet
6
Evaluating Limits Algebraically – Rationalizing Evaluate:
1. lim$→2
𝑥 − 1𝑥 − 1
2. lim$→3
2𝑥 + 1 − 3𝑥& − 16
3. lim
$→1&J𝑥& + 3𝑥 + 26 − 2 5 − 2𝑥
4. lim
$→K
2𝑥 − 12𝑥 − 2 − 10 − 𝑥
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Limits Evaluating Limits Algebraically – Common Denominators
Worksheet
7
Evaluating Limits Algebraically – Rationalizing Evaluate:
1. lim$→2
𝑥 − 1𝑥 + 3 − 5 − 𝑥
2.
lim$→.
𝑥/ − 2𝑥 − 8
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Limits Evaluating Limits Algebraically – Common Denominators
Worksheet
8
Evaluating Limits Algebraically – Common Denominators Evaluate:
1. lim$→1G
𝑥 + 71𝑥 +
17
2. lim$→2/&
2 + 1𝑥 − 1
2𝑥 − 1
3. lim$→:
8𝑥 + 1 +
4𝑥 − 5
𝑥& − 9
4. lim$→2
1𝑥 − 1
1𝑥 + 2
−2
3𝑥 + 3
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Limits Evaluating Limits Algebraically – Common Denominators
Worksheet
9
Evaluating Limits Algebraically – Common Denominators Evaluate:
1. lim$→2
1𝑥& − 1
1𝑥& − 3𝑥 + 2 −
1𝑥& − 2𝑥 + 1
2. lim
$→1&
𝑥 + 22𝑥 + 6
− 1𝑥 + 3
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Limits Evaluating Limits Algebraically – Quiz I
Worksheet
10
Evaluating Limits Algebraically – Quiz I Evaluate:
1. lim$→:
𝑥& − 9𝑥& − 2𝑥 − 2
2.
lim$→2
𝑥3 − 1𝑥: − 1
3. lim$→&
𝑥 − 2𝑥 + 2 − 2
4. lim$→12
𝑥 − 1𝑥 + 5 − 3
5. lim$→>
1𝑥 − 3 +
1𝑥 + 2
𝑥
6. lim$→1&
1𝑥 + 2
+5
𝑥& − 𝑥 − 6