Basic Calculus exercises

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) lim f (x) =
x→0−

(e) lim f (x) =
x→0

(f) lim f (x) =
x→3+

(g) lim f (x) =
x→3

(h) 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) lim f (x) =
x→−1

(e) lim f (x) =
x→0

(f) lim f (x) =
x→2+

(g) lim f (x) =
x→∞

,3. Evaluate each limit using algebraic techniques.
Use ∞, −∞ or DN E where appropriate.
x2 − 25
(a) lim x4 − 10
x→0 x2 − 4x − 5 (q) lim
x→∞ 4x3 + x
x2 − 25 r
(b) lim x−3
x→5 x2 − 4x − 5 (r) lim 3
x→−∞ 5−x
2
7x − 4x − 3
(c) lim 3x3 + x2 − 2
x→1 3x2 − 4x + 1 (s) lim
x→∞ x2 + x − 2x3 + 1
x4 + 5x3 + 6x2
(d) lim 2 x+5
x→−2 x (x + 1) − 4(x + 1) (t) lim
x→∞ 2x2 + 1
3
x5 + 1
 
(e) lim |x + 1| +
x→−3 x (u) lim cos
√ x→−∞ x6 + x5 + 100
x+1−2
(f) lim 2x
x→3 x2 − 9 (v) lim
√ x→2 x2−4
x2 + 7 − 3 3x
(g) lim (w) lim
x→3 x+3 x→−1 x2 + 2x + 1
x2 + 2x − 8 x2 − 25
(h) lim √ (x) lim
x→2 x2 + 5 − (x + 1) x→−1 x2 − 4x − 5
 2 1/3 √
2y + 2y + 4 x2 − 5 + 2
(i) lim (y) lim
y→5 6y − 3 x→3 x−3

(j)
p
lim 4 2 cos(x) − 5 2x + sin(x)
x→0
(z) lim
x→0 x4
1 1 1 2
− (A) lim− + ex
(k) lim 3 + x 3 − x x→1 x − 1
x→0 x
(B) lim 2x2 − 3x
2x + 8 1 x→∞
2
− √ √
(l) lim x − 12 x x+2− 2−x
x→−6 x+6 (C) lim
√ √
x→0 x
(m) lim x2 − 2 − x2 + 1 ex
x→∞ (D) lim+
√ √ x→0 1 + ln(x)
(n) lim x−2− x √
x→−∞
(E) lim x2 + 1 − 2x
√
6
x→∞
(o) lim 2x − 14 √
x→7 3
x−1
√ (F) lim √
(p) lim− 3 − 3x x→1 x−1
x→1

,4. Find the following limits involving absolute values.

x2 − 1 1 x2 |x − 3|
(a) lim (b) lim + x2 (c) lim−
x→1 |x − 1| x→−2 |x + 2| x→3 x−3


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) = sin(πx) for x < 1,
x→0 f (x) = 2
2x for x > 1.
(c) lim f (x) =
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) = 
 t2 for t < −2
(d) lim f (t) =


 t+6
t→−2
f (t) = for − 1 < t < 2
(e) lim f (t) =

 t2 − t

t→−1+ 
3t − 2 for t ≥ 2
(f) lim f (t) =
t→2

(g) lim f (t) =
t→0

, ANSWERS:
1. (a) DNE (b) 0 (c) 3 (d) −∞ (e) DNE (f) 2 (g) DNE (h) 1
2. (a) 0 (b) DNE (c) 0 (d) DNE (e) 0 (f) −∞ (g) 1
3.

(a) 5 1
(l) 36 (w) −∞
5
(b) 3 (m) 0
(x) DNE
(c) 5 (n) DNE
(y) DNE
(d) 1 (o) DNE
(z) ∞
(e) 1 (p) 0
(A) −∞
1
(f) 24 (q) ∞
(B) ∞
1
(g) (r) −1
6 (C) √1
2
(h) −18 (s) − 32
4
(D) 0
(i) 3 (t) 0
(E) −∞
(j) DNE (u) 1 2
(F) 3
(k) − 29 (v) DNE

4. (a) DNE (b) ∞ (c) −9
5. k = −1, limit is then equal to 9
6. (a) DNE (b) 0 (c) DNE
5
7. (a) DNE (b) 4 (c) 10 (d) DNE (e) 2 (f) 4 (g) DNE
5
8. (a) 0 (b) 0 (c) 3