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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