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
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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.
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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
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Worksheet by Kuta Software LLC