Kuta Software - Infinite Calculus

Name___________________________________

Evaluating Limits

Date________________ Period____

Evaluate each limit. 1) lim+ x → −1

4x + 4

2) lim− f ( x) , f ( x) =

x+1

x → −1

8

4

6

2

4

−8

−6

−4

−4

2

2

4

6

6

x

−4

x

−2

−6

−4

−8

−6

−10

−8

−12

x → −3

4

−2

−2

3) lim f ( x) , f ( x) =

− x 2 − 4 x − 4, x > −1

−2

2 −6

x ≤ −1

f(x)

f(x)

−8

{

− x − 8,

− x 2 − 10 x − 24, x ≤ −3

{

2 x + 3,

x > −3

4) lim f ( x) , f ( x) = x → −1

f(x)

{

x < −1

x, 2

− x + 2 x, x ≥ −1

f(x) 6

4 4 2 2 −10

−8

−6

−4

−2

2

4

x −8

−2

−6

−4

−2

2

4

6

x

−2 −4 −4 −6 −6 −8 −8 −10

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−10

-1-

Worksheet by Kuta Software LLC

Evaluate each limit. You may use the provided graph to sketch the function. 5) lim− f ( x) , f ( x) = x → −1

{

− x − 3, x ≤ −1 x + 1, x > −1

6) lim f ( x) , f ( x) = x → −2

− x 2 − 4 x − 5, x ≤ −2

{

f(x)

−8

−6

−4

f(x)

6

6

4

4

2

2

−2

x > −2

−1,

2

4

6

−10

x

−8

−6

−4

−2

2

−2

−2

−4

−4

−6

−6

−8

−8

4

6 x

Evaluate each limit. 7) lim+ f ( x) , f ( x) = x→0

{

1,

x≤0 2

− x + 4 x − 3, x > 0

9) lim+ −2 x + 1 x→0

8) lim− x→0

x → −1

x

10) lim f ( x) , f ( x) = x→1

11) lim

x

{

3 x+1 x+1

12) lim f ( x) , f ( x) = x → −2

x 9 + , 2 2

x<1

x 2 − 6 x + 10, x ≥ 1

{

x2 , −

x ≤ −2

x + 3, x > −2 2

Critical thinking questions: 13) Give an example of a two-sided limit of a piecewise function where the limit does not exist.

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14) Given an example of a two-sided limit of a function with an absolute value where the limit does not exist.

-2-

Worksheet by Kuta Software LLC

Kuta Software - Infinite Calculus

Name___________________________________

Evaluating Limits

Date________________ Period____

Evaluate each limit. 1) lim+ x → −1

4x + 4

2) lim− f ( x) , f ( x) =

x+1

x → −1

8

4

6

2

4

−8

−6

−4

−4

− x 2 − 4 x − 4, x > −1

−2

2

4

6

x

−2

2 −6

x ≤ −1

f(x)

f(x)

−8

{

− x − 8,

−2

2

4

6

−4

x

−2

−6

−4

−8

−6

−10

−8

−12

−7

4

3) lim f ( x) , f ( x) = x → −3

− x 2 − 10 x − 24, x ≤ −3

{

2 x + 3,

x > −3

4) lim f ( x) , f ( x) = x → −1

f(x)

{

x < −1

x, 2

− x + 2 x, x ≥ −1

f(x) 6

4 4 2 2 −10

−8

−6

−4

−2

2

4

x −8

−2

−6

−4

−2

2

4

6

x

−2 −4 −4 −6 −6 −8 −8 −10 −10

Does not exist.

−3

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-1-

Worksheet by Kuta Software LLC

Evaluate each limit. You may use the provided graph to sketch the function. 5) lim− f ( x) , f ( x) = x → −1

{

− x − 3, x ≤ −1

6) lim f ( x) , f ( x) =

x + 1, x > −1

x → −2

− x 2 − 4 x − 5, x ≤ −2

{

f(x)

−8

−6

−4

f(x)

6

6

4

4

2

2

−2

x > −2

−1,

2

4

6

−10

x

−8

−6

−4

−2

2

−2

−2

−4

−4

−6

−6

−8

−8

−2

4

6 x

−1

Evaluate each limit. 7) lim+ f ( x) , f ( x) = x→0

{

1,

x≤0

8) lim−

2

− x + 4 x − 3, x > 0

x→0

−3

x x

−1

9) lim+ −2 x + 1 x→0

10) lim f ( x) , f ( x) = x→1

0

{

x 9 + , 2 2

x<1

x 2 − 6 x + 10, x ≥ 1

5

11) lim

x → −1

3 x+1 12) lim f ( x) , f ( x) =

x+1

x → −2

Does not exist.

{

x2 , −

x ≤ −2

x + 3, x > −2 2

4 Critical thinking questions: 13) Give an example of a two-sided limit of a piecewise function where the limit does not exist. Many answers. Ex: lim f ( x) , f ( x) = x→1

{

14) Given an example of a two-sided limit of a function with an absolute value where the limit does not exist.

0, x < 1

Many answers. Ex: lim x→0

x, x ≥ 1

x x

Create your own worksheets like this one with Infinite Calculus. Free trial available at KutaSoftware.com

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-2-

Worksheet by Kuta Software LLC

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