INFINITE LIMIT WORKSHEET
MATH 141 – PROF. L. FINOTTI
Questions
1) Consider the graph of f(x) given below and compute the limits:
-3 -2 -1 1 2 3 4
-15
-10
-5
5
10
15
(a) lim
x→−1−f(x) =
(b) lim
x→−1+f(x) =
(c) lim
x→2−f(x) =
(d) lim
x→2+f(x) =
1
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INFINITE LIMIT WORKSHEET. MATH 141 – PROF. L. FINOTTI. Questions. 1) Consider the graph of f(x) given below and compute the limits:.
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INFINITE LIMIT WORKSHEET
MATH 141 – PROF. L. FINOTTI
Questions
- Consider the graph of f (x) given below and compute the limits:
-3 -2 -1 1 2 3 4
5
10
15
(a) lim x→− 1 −^
f (x) =
(b) lim x→− 1 +^
f (x) =
(c) lim x→ 2 −^
f (x) =
(d) lim x→ 2 +^
f (x) =
1
2 MATH 141 – PROF. L. FINOTTI
- Consider the graph of f (x) given below:
-3 -2 -1 1 2 3
5
10
15
At what values of x does f (x) has an infinite limit [as x approaches this value]? Write down
the side limits [both!] for each one of these values.
- Compute the following limits:
(a) lim x→ 2 ±
x + 1
x − 2
[The “±” here means that you have to compute both side limits individually.]
(b) lim x→ 3 ±
x 2 − 3 x − 1
x − 3
(c) lim x→ 0 ±
cos(x 2 − x)
x^2
(d) lim x→ 2 ±
e x− 2
x − 2
(e) lim x→ 1 ±
x 2 − 4 x + 3
x 2 − 2 x + 1
(f) lim x→π±
cos(2x)
sin(x)
4 MATH 141 – PROF. L. FINOTTI
Answers (and Hints)
1 : (a): ∞, (b): ∞, (c): −∞, (d): ∞.
2 : We have infinite limits for x = − 2 , 0 , 1. The side limits are:
lim x→− 2 −^
f (x) = ∞; lim x→− 2 +^
f (x) = −∞; lim x→ 0 −^
f (x) = ∞;
lim x→ 0 +^
f (x) = −∞; lim x→ 1 −^
f (x) = −∞; lim x→ 1 +^
f (x) = ∞.
3: We right first the limit from the left and the from the right in the answers.
(a) −∞, ∞
(b) ∞, −∞
(c) ∞, ∞
(d) −∞, ∞
(e) ∞, −∞. Hint:
x 2 − 4 x + 3
x^2 − 2 x + 1
(x − 1)(x − 3)
(x − 1)^2
(f) ∞, −∞
(g) −∞. Hint: lim x→ 0 +^
ln(x) = −∞.
(h) ∞. Hint:
x
sin(x)
x/x
sin(x)/x
x
sin(x)/x
(i) −∞, ∞, Hint:
sin(x)
1 − cos(x)
sin(x)/x
(1 − cos(x))/x