201-103-RE - Calculus 1
WORKSHEET: LIMITS
1. Use the graph of the function f (x) to answer each question.
Use ∞, −∞ or DN E where appropriate.
(a)
f (0) =
(b)
f (2) =
(c)
f (3) =
(d)
(e)
(f)
(g)
(h)
lim f (x) =
x→0−
lim f (x) =
x→0
lim f (x) =
x→3+
lim f (x) =
x→3
lim f (x) =
x→−∞
2. Use the graph of the function f (x) to answer each question.
Use ∞, −∞ or DN E where appropriate.
(a)
f (0) =
(b)
f (2) =
(c)
f (3) =
(d)
(e)
(f)
(g)
lim f (x) =
x→−1
lim f (x) =
x→0
lim f (x) =
x→2+
lim f (x) =
x→∞
3. Evaluate each limit using algebraic techniques.
Use ∞, −∞ or DN E where appropriate.
(a)
(b)
lim
x2 − 25
x2 − 4x − 5
(q)
lim
x2 − 25
x2 − 4x − 5
(r)
x→0
x→5
2
(c)
(d)
(e)
7x − 4x − 3
3x2 − 4x + 1
lim
x→1
x4 + 5x3 + 6x2
lim 2
x→−2 x (x + 1) − 4(x + 1)
lim |x + 1| +
x→−3
√
(f)
lim
x→3
√
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
(s)
lim
x→3
3
x
(u)
x+1−2
x2 − 9
(v)
x2 + 7 − 3
x+3
(w)
x2 + 2x − 8
x→2
x2 + 5 − (x + 1)
2
1/3
2y + 2y + 4
lim
y→5
6y − 3
p
lim 4 2 cos(x) − 5
lim √
x→0
1
1
−
lim 3 + x 3 − x
x→0
x
1
2x + 8
−
2
lim x − 12 x
x→−6
x+6
√
√
lim x2 − 2 − x2 + 1
x→∞
√
lim
x→−∞
lim
√
6
x→7
lim−
x→1
(t)
x−2−
√
(x)
(y)
(z)
(A)
(B)
(C)
(D)
x
2x − 14
3 − 3x
lim
x→∞
3x3 + x2 − 2
x2 + x − 2x3 + 1
x+5
x→∞ 2x2 + 1
x5 + 1
lim cos
x→−∞
x6 + x5 + 100
lim
lim
x→2
2x
−4
x2
lim
x→−1 x2
3x
+ 2x + 1
x2 − 25
x→−1 x2 − 4x − 5
√
x2 − 5 + 2
lim
x→3
x−3
lim
2x + sin(x)
x→0
x4
1
2
lim−
+ ex
x→1 x − 1
lim
lim 2x2 − 3x
x→∞
√
(E)
√
x4 − 10
x→∞ 4x3 + x
r
x−3
lim 3
x→−∞
5−x
lim
(F)
√
x+2− 2−x
lim
x→0
x
ex
lim+
x→0 1 + ln(x)
√
lim x2 + 1 − 2x
x→∞
√
3
x−1
lim √
x→1
x−1
4. Find the following limits involving absolute values.
x2 − 1
(a) lim
x→1 |x − 1|
x2 |x − 3|
(c) lim−
x→3
x−3
1
(b) lim
+ x2
x→−2 |x + 2|
5. Find the value of the parameter k to make the following limit exist and be finite.
What is then the value of the limit?
x2 + kx − 20
lim
x→5
x−5
6. Answer the following questions for the piecewise defined function f (x) described on
the right hand side.
(a)
f (1) =
(b)
lim f (x) =
(c)
x→0
(
sin(πx)
f (x) =
lim f (x) =
2x
2
for x < 1,
for x > 1.
x→1
7. Answer the following questions for the piecewise defined function f (t) described on
the right hand side.
(a)
f (−3/2) =
(b)
f (2) =
(c)
f (3/2) =
(d)
lim f (t) =
(e)
t→−2
lim f (t) =
t→−1+
(f)
lim f (t) =
(g)
lim f (t) =
t→2
t→0

t2



 t+6
f (t) =

t2 − t



3t − 2
for t < −2
for − 1 < t < 2
for t ≥ 2
ANSWERS:
(b) 0
(c) 3
(d) −∞
(b) DNE
(c) 0
(d) DNE
1. (a) DNE
2. (a) 0
(e) DNE
(f) 2
(f) −∞
(e) 0
(g) DNE
(h) 1
(g) 1
3.
(a) 5
(b)
(l)
5
3
1
36
(w) −∞
(m) 0
(x) DNE
(c) 5
(n) DNE
(d) 1
(o) DNE
(e) 1
(p) 0
(f)
(g)
1
24
1
6
(B) ∞
(s)
(j) DNE
(u) 1
− 29
(C)
− 32
(t) 0
4. (a) DNE
(A) −∞
(r) −1
4
3
(k)
(z) ∞
(q) ∞
(h) −18
(i)
(y) DNE
(D) 0
(E) −∞
(F)
(v) DNE
(b) ∞
(c) −9
5. k = −1, limit is then equal to 9
6. (a) DNE
(b) 0
(c) DNE
7. (a) DNE
(b) 4
(c) 10
8. (a) 0
(b) 0
(c)
5
3
(d) DNE
√1
2
(e)
5
2
(f) 4
(g) DNE
2
3