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Worksheet by Kuta Software LLC
Kuta Software - Infinite Calculus Name___________________________________
Period____Date________________
Evaluating Limits
Evaluate each limit.
1)
lim
x
→
2
f
(
x
)
,
f
(
x
)
=
{
−
x
2
+ 2,
x ≠ 2
−5,
x = 2
x
f(x)
−6 −4 −2 2 4 6 8 10
−10
−8
−6
−4
−2
2
4
2)
lim
x
→
−2
−
x
2
− 4
x + 2
x
f(x)
−10 −8 −6 −4 −2 2 4 6
−4
−2
2
4
6
8
10
12
Evaluate each limit. You may use the provided graph to sketch the function.
3)
lim
x
→
3
x
2
− 7
x + 12
x − 3
x
f(x)
−4 −2 2 4 6 8 10
−8
−6
−4
−2
2
4
6
4)
lim
x
→
−3
x + 3
x
2
+ 2
x − 3
x
f(x)
−10 −8 −6 −4 −2 2 4
−8
−6
−4
−2
2
4
6
8
Evaluate each limit.
5)
lim
x
→
0
f
(
x
)
,
f
(
x
)
=
{
x + 1,
x ≠ 0
2,
x = 0
6)
lim
x
→
3
f
(
x
)
,
f
(
x
)
=
{
2 +
x
2
,
x ≠ 3
2,
x = 3
-1-
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Worksheet by Kuta Software LLC
7)
lim
x
→
1
−
x
2
− 1
x − 1
8)
lim
x
→
5
−
x
2
− 5
x
x − 5
9)
lim
x
→
2
−
x
2
−
x − 2
x − 2
10)
lim
x
→
−5
x
2
+ 3
x − 10
x + 5
11)
lim
x
→
0
1
−4 +
x
+
1
4
x
12)
lim
x
→
−3
x
1
3 +
x
−
1
3
13)
lim
x
→
5
x −
5
x + 4 −
3
14)
lim
x
→
3
x + 6 −
3
x −
3
Critical thinking questions:
15) Give an example of a limit of a rational function
where the limit at -1 exists, but the rational
function is undefined at -1.
16) Give two values of
a where the limit cannot be
solved using direct evaluation. Give one value
of
a where the limit can be solved using direct
evaluation.
lim
x
→
a
x
1
−2 +
x
+
1
2
-2-
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Calculus Name___________________________________
Period____Date________________
Evaluating Limits
Evaluate each limit.
1)
lim
x
→
2
f
(
x
)
,
f
(
x
)
=
{
−
x
2
+ 2,
x ≠ 2
−5,
x = 2
x
f(x)
−6 −4 −2 2 4 6 8 10
−10
−8
−6
−4
−2
2
4
−2
2)
lim
x
→
−2
−
x
2
− 4
x + 2
x
f(x)
−10 −8 −6 −4 −2 2 4 6
−4
−2
2
4
6
8
10
12
4
Evaluate each limit. You may use the provided graph to sketch the function.
3)
lim
x
→
3
x
2
− 7
x + 12
x − 3
x
f(x)
−4 −2 2 4 6 8 10
−8
−6
−4
−2
2
4
6
−1
4)
lim
x
→
−3
x + 3
x
2
+ 2
x − 3
x
f(x)
−10 −8 −6 −4 −2 2 4
−8
−6
−4
−2
2
4
6
8
−
1
4
Evaluate each limit.
5)
lim
x
→
0
f
(
x
)
,
f
(
x
)
=
{
x + 1,
x ≠ 0
2,
x = 0
1
6)
lim
x
→
3
f
(
x
)
,
f
(
x
)
=
{
2 +
x
2
,
x ≠ 3
2,
x = 3
7
2
-1-
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Worksheet by Kuta Software LLC
7)
lim
x
→
1
−
x
2
− 1
x − 1
−2
8)
lim
x
→
5
−
x
2
− 5
x
x − 5
−5
9)
lim
x
→
2
−
x
2
−
x − 2
x − 2
−3
10)
lim
x
→
−5
x
2
+ 3
x − 10
x + 5
−7
11)
lim
x
→
0
1
−4 +
x
+
1
4
x
−
1
16
12)
lim
x
→
−3
x
1
3 +
x
−
1
3
0
13)
lim
x
→
5
x −
5
x + 4 −
3
6
14)
lim
x
→
3
x + 6 −
3
x −
3
1
6
Critical thinking questions:
15) Give an example of a limit of a rational function
where the limit at -1 exists, but the rational
function is undefined at -1.
Many answers. Ex:
lim
x
→
−1
x
2
− 1
x + 1
16) Give two values of
a where the limit cannot be
solved using direct evaluation. Give one value
of
a where the limit can be solved using direct
evaluation.
lim
x
→
a
x
1
−2 +
x
+
1
2
No direct eval:
a=0,2 Direct eval:
a=any other number
-2-
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