1. The graph of a function f is given above, answer the questions:
a. f (−5) =? Ans: 2
b. lim f (x) = ? Ans: 2
c.
f.
lim f (x) = ? Ans: 1
g.
lim f (x) = ? Ans: 3
k.
x→−5−
lim f (x) =? Ans: 2
d. lim f (x) =? Ans:
lim f (x) =? Ans: 1
h. lim f (x) =? Ans:
lim f (x) =? Ans: 3
l. lim f (x) =? Ans:
x→−5
x→−5+
2
e. f (−4) =? Ans: 2
x→−4−
x→−4
x→−4+
1
i. f (−3) =? Ans: 3
j.
x→−3−
x→−3
x→−3+
3
m. f (−2) =? Ans: 4
n.
lim f (x) = ? Ans: 5
o.
x→−2−
lim f (x) =? Ans: 4
p.
x→−2+
lim f (x) =?
x→−2
Ans: Does not exist
q. f (−1) =? Ans: 2
r.
lim f (x) = ? Ans: 2
x→−1−
s.
lim f (x) =? Ans: 2
t. lim f (x) =? Ans:
x→−1
x→−1+
2
u. f (0) =? Ans: Undefined.
v. lim− f (x) = ? Ans: 4
w. lim+ f (x) =? Ans: 2
x→0
x. lim f (x) =?
x→0
x→0
Ans: Does not exist
y. f (1) =? Ans: 1
z. lim f (x) = ? Ans: 1
cc. f (2) =? Ans: 0
x→1−
dd. lim− f (x) = ? Ans: 0
x→2
aa. lim f (x) =? Ans: 1
x→1+
ee. lim+ f (x) =? Ans: -2
x→2
bb. lim f (x) =? Ans: 1
x→1
ff. lim f (x) =? Ans:
x→2
Does not exist
gg. f (3) =? Ans: ≈ −1.8
hh. lim− f (x) = ? Ans: ≈ −1.8
x→3
lim f (x) =? Ans: ≈ −1.8
x→3
ii. lim+ f (x) =? Ans: ≈ −1.8
x→3
jj.
kk. f (4) =? Ans: Undefined
ll.
lim f (x) = ? Ans: -1
x→4−
mm.
lim f (x) =? Ans: ∞
nn.
lim f (x) =? Ans: ≈ 1.5
rr.
x→4+
lim f (x) =? Ans: Does not exist
x→4
oo. f (5) =? Ans: ≈ 1.5
lim f (x) =? Ans: ≈ 1.5
x→5
pp.
lim f (x) = ? Ans: ≈ 1.5
x→5−
qq.
x→5+
2. The graph of a function f is given above, answer the questions:
a. f (−5) =? Ans: 1
b. lim f (x) = ? Ans: 1
x→−5−
c.
lim f (x) =? Ans: 1
d. lim f (x) =? Ans:
x→−5
x→−5+
1
e. f (−4) =? Ans: 1
f.
lim f (x) = ? Ans: −1
g.
x→−4−
lim f (x) =? Ans: 1
h.
x→−4+
lim f (x) =?
x→−4
Ans: Does not exist.
i. f (−3) =? Ans: 0
j.
lim f (x) = ? Ans: 0
x→−3−
k.
lim f (x) =? Ans: 3
l. lim f (x) =? Ans:
x→−3
x→−3+
Does not exist.
m. f (−2) =? Ans: 2
n.
lim f (x) = ? Ans: 2
x→−2−
o.
lim f (x) =? Ans: 2
p.
x→−2+
lim f (x) =?
x→−2
Ans: 2
q. f (−1) =? Ans: Undefined.
r.
lim f (x) = ? Ans: 1
s.
x→−1−
lim f (x) =? Ans: −∞
t.
lim f (x) =? Ans: ≈ 0.3
x.
x→−1+
lim f (x) =? Ans: Does not exist.
x→−1
u. f (0) =? Ans: ≈ 0.3
v.
lim f (x) = ? Ans: ≈ 0.3
w.
x→0−
x→0+
lim f (x) =? Ans: ≈ 0.3
x→0
y. f (1) =? Ans: −1
z. lim f (x) = ? Ans: 1
x→1−
aa. lim f (x) =? Ans: 0
bb. lim f (x) =? Ans:
x→1
x→1+
Does not exist.
cc. f (2) =? Ans: −2
dd.
lim f (x) = ? Ans: 1
x→2−
ee. lim+ f (x) =? Ans: −2
ff. lim f (x) =?
x→2
x→2
Ans: Does not exist
gg. f (3) =? Ans: 0
hh. lim− f (x) = ? Ans: 0
x→3
ii. lim+ f (x) =? Ans: 0
x→3
jj. lim f (x) =? Ans: 0
x→3
kk. f (4) =? Ans: −1
ll. lim− f (x) = ? Ans: 2
x→4
mm. lim+ f (x) =? Ans: −1
nn. lim f (x) =?
x→4
x→4
Ans: Does not exist.
oo. f (5) =? Ans: 0
pp. lim− f (x) = ? Ans: 0
x→5
0
qq. lim+ f (x) =? Ans: 0
x→5
rr. lim f (x) =? Ans:
x→5
3. Evaluate the limit:
a. lim 4x + 3
x→1
Ans. lim 4x + 3 = 7
x→1
x2 − 1
x→1 x − 1
b. lim
x2 − 1
=2
x→1 x − 1
Ans. lim
x3 − 1
x→1 x − 1
c. lim
x3 − 1
=3
x→1 x − 1
Ans. lim
x−1
x→1 x2 − 1
1
x−1
=
Ans. lim 2
x→1 x − 1
2
d. lim
x−1
x2 − 2x + 1
x−1
Ans. lim 2
= DNE
x→1 x − 2x + 1
e. lim
x→1
x2 − 5x
x→5 x2 + 2x − 35
f. lim
Ans. lim
x→5 x2
5
x2 − 5x
=
+ 2x − 35
12
√
x+2−3
x−7
√
x+2−3
1
=
Ans: lim
x→7
x−7
6
g. lim
x→7
sin x
x
sin x
Ans. lim
=1
x→0 x
h. lim
x→0
f (x) − f (2)
x→2
x−2
i. lim
Ans. lim
x→2
f (x) − f (2)
=3
x−2
f (x) − f (3)
x→3
x−3
j. lim
Ans. lim
x→3
Ans. lim
x→c
if f (x) = x2 − 1
f (x) − f (3)
=6
x−3
f (x) − f (c)
x→c
x−c
k. lim
if f (x) = 3x + 1
if f (x) = 2x2
f (x) − f (c)
= 4c
x−c
f (x + h) − f (x)
h→0
h
l. lim
if f (x) = −4x − 2
f (x + h) − f (x)
= −4
h→0
h
Ans: lim
f (x + h) − f (x)
h
m. lim
h→0
if f (x) = sin x
f (x + h) − f (x)
= cos x
h→0
h
Ans: lim
n. lim
h→0
f (x + h) − f (x)
h
if f (x) = cos x
f (x + h) − f (x)
= − sin x
h→0
h
Ans: lim
o. lim−
x→2
|x − 2|
x−2
Ans: −1
p. lim+
x→2
|x − 2|
x−2
Ans: 1
q.
lim −
x→−1
x
x+1
Ans: ∞
r.
lim
x→−1+
x
x+1
Ans: −∞
4. Let
Evaluate:
a.
lim f (x)
x→−3−
Ans: 10
b.
lim f (x)
x→−3+
Ans: −5
c. lim f (x)
x→−3
Ans: Limit Does Not Exist.
d.
lim f (x)
x→−1−
Ans: −3
e.
lim f (x)
x→−1+
Ans: −3
f. lim f (x)
x→−1
Ans: −3
g. lim− f (x)
x→0
 2
 x +1
x−
f (x) =
√2

x
if x < −3
if − 3 ≤ x < 9
if 9 ≤ x
Ans: −2
h. lim+ f (x)
x→0
Ans: −2
i. lim f (x)
x→0
Ans: −2
j. lim f (x)
x→2−
Ans: 0
k. lim f (x)
x→2+
Ans: 0
l. lim f (x)
x→2
Ans: 0
m. lim− f (x)
x→9
Ans: 7
n. lim+ f (x)
x→9
Ans: 3
o. lim f (x)
x→9
Ans: Limit Does Not Exist
p.
lim f (x)
x→16−
Ans: 4
q.
lim f (x)
x→16+
Ans: 4
r. lim f (x)
x→16
Ans: 4
5. For the given function f , find the point(s) where f is discontinuous. Catagorize the point(s) you found as
removable or unremovable discontinuity. If the point is a removable discontinuity, explain how you can redefine
the function at that point to make the function continuous at that point.
a. f (x) = 5
Ans: f is continuous on whole real line.
b. f (x) = x3 + 2x + 7
Ans: f is continuous on whole real line.
c. f (x) = |x|
Ans: f is continuous on whole real line.
sin x
x
Ans: f is discontinuous at x = 0. Define f (x) = 1 if x = 0 and f will be continuous at 0, so x = 0 is a removable
d. f (x) =
discontinuity.
x2 − 3x + 2
x2 − 4
e. f (x) =
1
if x = 2, and f will be continuous at 2, so x = 2
4
is a removable discontinuity. lim f (x) does not exist, so x = −2 is a non-removable discontinuity.
Ans: f is discountinuous at x = −2 and x = 2. Define f (x) =
x→−2
|x + 4|
x+4
f. f (x) =
Ans: f is discontinuous at x = −4. This is a non-removable discontinuity.
2
g. f (x) = e−1/x
Ans: f is discontinuous at x = 0. Define f (x) = 0 if x = 0 will make f continuous at 0, so x = 0 is a removable
discontinuity.
h. f (x) =


x+3
x2 + 1

−2x + 10
if x < −1
if − 1 ≤ x < 4
if 4 ≤ x
Ans: f has a non-removable discontinuity at x = 4.
i. f (x) =


2
x−1

2x − 3
if x ≤ 3
if 3 ≤ x < 5
if 5 ≤ x
Ans: f has a non-removable discontinuity at x = 5.
6. Using the same graph of the function at problem (1), answer the following question:
a. For which value(s) of x does f have a removable discontinuity?
Ans: x = −4
b. For which value(s) of x does f have a non-removable discontiuity?
Ans: x = −2, x = 0, x = 2, x = 4
c. For which value(s) of x is f discontinuous but has a limit?
Ans: x = −4
d. For which value(s) of x is f discontinuous and not have a limit, but has a left and/or right hand limit?
Ans: x = −2, x = 0, x = 2, x = 4
e. For which value(s) of x does f not have any left or right hand limit?
Ans: None.
7. Using the same graph of the function in problem (2), answer the following question:
a. For which value(s) of x does f have a removable discontinuity?
Ans: None.
b. For which value(s) of x does f have a non-removable discontiuity?
Ans: x = −4, x = −3, x = −1, x = 1, x = 2, x = 4
c. For which value(s) of x is f discontinuous but has a limit?
Ans: None.
d. For which value(s) of x is f discontinuous and not have a limit, but has a left and/or right hand limit?
Ans: x = −4, x = −3, x = −1, x = 1, x = 2, x = 4
e. For which value(s) of x does f not have any left or right hand limit?
Ans: None.
8. Find the slope of the tangent line to the given function at the give value of x, using the limit definition.
x = −2
a. f (x) = 3,
Ans: m = 0
b. f (x) = 3x + 1,
x=1
Ans: m = 3
c. f (x) = x2 − 1,
Ans: m = 6
√
d. f (x) = x,
Ans: m =
x=3
x=9
1
6
9. For the functions in (8), find the equation of the tangent line to the given function at the give value of x.
a. Ans: y = 3
b. Ans: y = 3x + 1
c. Ans: y = 6x − 10
d. Ans: y =
1
3
x+
6
2