2
Limits
Sheet 2. Limits
Exercise 2.1.
Find the limits (if exist):
a) lim (x2 + 5x − 6)
b) lim (−2x7 + 3x2 − 4)
x→∞
x→∞
2
x + 3x
x→∞ x2 − 1
x2 − 2
g) lim
x→∞
x
x−1
j) lim
x→∞ −x2 + 2x − 1
(1 − 2x)3
m) lim
x→∞ (2x + 3)2 (1 − 7x)
d) lim
2x + 3x
x→∞ 3x3 + x + 1
x5 + 2x2 + 5x
h) lim
x→∞
−2x3 + 1
x3 + 2x − 1
k) lim
x→∞
3x4 + x
1 − 2x
√
n) lim
x→∞ 2 +
x
e) lim
x→∞
2
2x + 3x − 7
x→∞ x2 + 4x − 2
t) lim
x+1
1 x2 −1
v) lim
x→∞
3
6x3 − 1
x→∞ 3x3 + 2x − 4
−3x3 + 1
i) lim
x→∞ −5x2 + 4
x2 + 2x − 1
l) lim 3
x→∞ x + 2x − 1
√
2+ x
o) lim
x→∞ 1 − 2x
f) lim
r) lim 3 −x2 +2
x→∞
x2 +1
x
2 5 +x
x→∞
x3 −1
q) lim 2 x+2
x→∞
x→−∞
2
x2
x
p) lim e x+1
s) lim
3
c) lim (2x5 − 3x5 + x2 − 1)
x3 − 2x2
x→−∞ 5x3 + x2 − x + 2
u) lim
x3 + 5x
w) lim
x→∞ x − 1
x) lim
y) lim 5
z) lim (−2x + 5x − 7)
x4 +5x−6
1
aa) lim
x→−∞
e
ab) lim (−2x6 + 5x − 4)
ac) lim 3x
x4 +2x2 +3
3
x→∞
x→∞
x→−∞
Exercise 2.2.
Find the limits:
a) lim (x2 + 5x − 6)
x→1
x→−∞
x2 + 1
x→3 x2 − 1
b) lim
3 +2x−7
x→−∞
x2 −1
e x+1
ad) lim (−2x5 + 6x4 − 3x + 7)
x→−∞
√
c) lim x x2 + 5
x→2
Find the limits:
x2 + 6x − 16
x2 − x − 2
a) lim
b) lim
x→2
x→−1
x−2
x+1
x2 cos x
x→0
3x
d) lim
Exercise 2.3.
x2 − 2x − 3
x→3
3−x
c) lim
x2 + 3x − 4
x→−4 x2 + 5x + 4
d) lim
Find the one-sided limits of f at the point x0 ,if:
1
1
1
, x0 = 3 b) f (x) =
, x0 = 3 c) f (x) =
, x0 = 3
a) f (x) =
x−3
3−x
(3 − x)2
1
x+1
1
d) f (x) =
, x0 = 1 e) f (x) = 2
, x0 = 2 f) f (x) = 2 x−1 , x0 = 1
x−1
x −4
1
1
x
g) f (x) = 4 x2 −4 , x0 = 2 h) f (x) = e 4−x2 , x0 = −2 i) f (x) =
1 , x0 = 0
1 + ex
Decide, if the two-sided limits exist.
Exercise 2.4.
Exercise 2.5.
Find the limit (one-sided or two-sided):
Last update: October 23, 2013
1
REFERENCES
References
a) lim
x+1
x→2 x − 2
x2 + x
e) lim
x→2 2 − x
b) lim
x2 + 4x + 3
x→1
x−1
2
x −1
f) lim
x→4 4 − x
c) lim
x2 + 2x − 3
x→−2
x+2
2
x −x−6
g) lim
x→1
1−x
x2 − 5x
x→3 x + 3
x2 − x − 2
h) lim1
1 − 2x
x→ 2
x+1
i) lim 2
x→1 x + 2x − 3
x2 − 2x − 3
j) lim
x→−2
x2 + 2x
x+3
k) lim
x→3 −x2 + 2x + 3
x2 + 3x − 10
l) lim
x→−1 −x2 − 5x − 4
Exercise 2.6.
Exercises 167 [1, pp. 7478].
Exercise 2.7.
Exercises 164 [1, pp. 8790].
Answers
d) lim
a) ∞, b) −∞, c) −∞, d) 1, e) 23 , f) 2, g) ∞, h) −∞, i) ∞, j) 0, k) 0, l) 0, m) 27 ,
n) −∞, o) 0, p) e, q) ∞, r) 0, s) 1, t) 2, u) 51 , v) 1, w) ∞, x) 0, y) ∞, z) −∞, aa) 0, ab) −∞, ac) 0,
ad) ∞.
5
Exercise 2.2. a) 0, b) , c) 6, d) 0.
4
5
Exercise 2.3. a) 10, b) −3, c) −4, d) .
3
Exercise 2.4. a) ∞, −∞, b) ∞, −∞, c) ∞, ∞, d) −∞, ∞, e) −∞, ∞, f) 0, ∞, g) 0, ∞, h) 0,
∞, i) 0, 0.
Exercise 2.5. a) does not exist, the one-sided limits are: −∞, ∞, b) does not exist, the one-sided
limits are: −∞, ∞, c) does not exist, the one-sided limits are: ∞, −∞, d) −1, e) does not exist, the
one-sided limits are: ∞, −∞, f) does not exist, the one-sided limits are: ∞, −∞, g) does not exist,
the one-sided limits are: −∞, ∞, h) does not exist, the one-sided limits are: −∞, ∞, i) does not
exist, the one-sided limits are: −∞, ∞, j) does not exist, the one-sided limits are: ∞, −∞, k) does
not exist, the one-sided limits are: ∞, −∞, l) does not exist, the one-sided limits are: −∞, ∞.
Exercise 2.1.
References
[1] Homann et al., Applied Calculus for Business, Economics, and
McGraw-Hill International Edition, Expanded eleventh edition.
Last update: October 23, 2013
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