1.3 Limits
KEY CONCEPTS
• A sequence is a function, f (n) = tn, whose domain is the set of natural numbers ℕ.
f (x) exists if the following three criteria are met:
• lim
xa
1. lim– f (x) exists
xa
2. lim+ f (x) exists
xa
3. lim– f (x) = lim+ f (x)
xa
xa
• lim
f (x) = L, which is read as “the limit of f (x), as x approaches a, is equal to L.”
xa
f (x) ≠ lim+ f (x), then lim
f (x) does not exist.
• If lim
xa
xa–
xa
• A function f (x) is continuous at a value x = a if the following three conditions
are satisfied:
1. f (a) is defined, that is, a is in the domain of f (x)
2. lim
f (x) exists
xa
3. lim
f (x) = f (a)
xa
b)
A
1. Each graph represents a sequence.
Determine the limit of each sequence, if it
exists.
a)
y
60
50
40
y
0.4
30
20
1 2 3 4 5 6 7 8 9x
⫺0.4
10
⫺0.8
⫺1.2
⫺1.4
1 2 3 4 5 6 7 8 9x
c)
y
8
6
4
2
1 2 3 4 5 6 7 8 9x
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MHR • Chapter 1 Rates of Change 978-0-07-073589-7
2. For each sequence, determine the limit or
explain why the limit does not exist.
a) 1, 2, 3, 2, 4, 2, 5, 2, 6, 2, …
b) –7.1, –7.01, –7.001, –7.001, –7.0001,
–7.000 001, …
7
3 4 5 6 __
,– ,…
c) – 2, – __, – __, – __, – __
2 3 4 5 6
1
1 , ___
d) 125, –25, 5, – __
,…
5 25
e) 9.9, 10.1, 9.99, 10.01, 9.999, 10.0001, …
B
7. Examine the given graph and evaluate the
following limits.
y
8
6
4
2
2
0
3. What is true about lim f(x) if
4
6
8
x
x3
lim– f(x) = –5 and lim+ f(x) = 5?
x3
x3
4. If lim– f (x) = 1.5, lim+ f(x) = 1.5, and
x–2
x–2
f(–2) = 1.5, what is true about lim f(x) ?
x–2
5. If lim– f(x) = – 2, lim+ f(x) = – 2 , and
x6
x6
f(6) = 0, what is true about lim f(x) ?
x6
6. A function y = f(x) exists such that
lim f(x) = 9, lim+ f(x) = 7, and f (–4) = 7.
x–4–
x–4
Is each statement is true or false? If the
statement is false, explain why.
a) y = f(x) is continuous at x = –4.
b) The limit of f(x) as x approaches –4
does not exist.
c) When x = –4, the value of the
function is 9.
d) The right-hand limit and the value of
the function are equal.
e) When x = –4, the value of the
function is 7.
f ) The function is discontinuous at
x = –4.
a) lim– f(x)
x5
b) lim+ f(x)
x5
c) lim f(x)
x5
8. Evaluate each limit, if it exists.
a) lim(2x2 + 4x – 5)
x–1
2x2 – 5
b) lim ______
x–3 x – 1
(
)
(
)
5
c) lim ______
x1 3x + 4
x2 – 4x + 3
d) lim __________
x–1
x2
(
)
9. A function y = f (x) exists such that
lim– f(x) = 3, lim+ f(x) = m, and f(1) = n.
x1
x–1
Determine the values of m and n that
make each statement true.
a) lim f(x) = 3, but y = f(x) is
x1
discontinuous at x = 1.
b) y = f(x) is continuous at x = 1.
c) lim f(x) does not exist.
x1
d) lim f(x) does not exist, but the
x1
right-hand limit and the value of
the function are equal.
1.3 Limits • MHR
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10. The general term of an infinite sequence
n
is defined by tn = 2 + _______.
(n + 1)
a) Write the first eight terms of the
sequence.
b) Explain why the limit of the sequence
is 3.
11. The general term of an infinite sequence
2(n + 1)
is defined by tn = 4 − ________
n .
a) Write the first ten terms of the
sequence.
b) Does the sequence have a limit?
Explain.
12. The general term of an infinite sequence
is defined by tn = 3n + n2.
a) Write the first ten terms of the
sequence.
b) Determine if the sequence has a
limit. Justify your answer.
13. Determine the fraction that is equivalent
to the limit of each sequence.
a) 0.6, 0.66, 0.666, 0.6666, …
b) 0.16, 0.166, 0.1666, 0.166 66, …
c) 0.09, 0.0909, 0.090 909, …
14. Consider these sequences.
3 4 5 6 __
7
n+2
, , … , ______
a) __, __, __, __
2 3 4 5 6
n+1
1 1 5 1 9
b) 6 __, 7 __, 6 __, 7__, 6 ___, … , 7 +
2 4 6 8 10
(–1)n
_____
,…
2n
9 25
n2
1 4 __
c) __, __
, , ___, … , ____
2 5 4 6
n+1
For each sequence,
i) Determine if the limit exists. If it
exists, state the limit. If it does not
exist, explain why. Use a graph to
support your answer.
15. Consider this graph.
y
4
2 0
C
4
2
8
12
16
y x4 8x2
20
a) What is the domain of the function?
b) Evaluate each limit.
i) lim– (x4 – 8x2)
x–2
ii) lim+ (x4 – 8x2)
x–2
iii) lim (x4 – 8x2)
x–2
c) Determine f(–2).
d) Is the graph continuous at x = –2?
Explain.
e) Is the graph continuous at x = 0 and
x = 2? Justify your answers.
16. City officials determine that the cost,
C, in dollars, of purifying water
can be represented by the function
60 000
C(p) = ______
p – 5000, where p is the
percent of impurities remaining in the
water after purification.
a) State the domain of this function.
b) Evaluate lim+C(p).
p0
c) Interpret the meaning of your answer
to part b).
d) Graph the function. How does
the graph support your result from
part b)?
ii) Write a limit expression to represent
the behaviour of the sequence.
12
x
MHR • Chapter 1 Rates of Change 978-0-07-073589-7
17. In a recursive sequence, each term is
defined in terms of the previous term.
Consider this recursive sequence:
1
t1 = 1, tn = ________, n ≥ 2.
1 + tn − 1
a) Determine the first eight terms of the
sequence.
21. Let f(x) =
b) Determine each limit, if it exists.
i) lim+ f (x)
x–1
ii) lim– f (x)
x–1
iii) lim f (x)
x–1
iv) lim f (x)
c) Write the sequence formed by taking
the reciprocal of each term of the
sequence in part a).
e) Compare your answers to parts b)
and d). What do you notice?
C
18. a) Determine the limit of the sequence
with general term tn = 2(−1)n −1.
b) Determine the limit of the sum of the
terms of the sequence from part a).
19. a) Write the first eight terms of__ the
sequence defined by tn = n√n , n ≥ 1.
b) Write a limit expression to represent
the behaviour of the sequence from
part a). Justify your answer.
20. Consider the sequence with general term
1
tn = _____
, n ≥ 1. Verify algebraically and
2n − 1
graphically that the limit of the sum of
this sequence is less than 2.
if x ≤ −1
if x > −1
a) Graph the function.
b) Write a limit expression to represent
the behaviour of the sequence from
part a).
d) Write a limit expression to represent
the behaviour of the sequence from
part c).
{
2 − x2
x −1
x0
v) lim f (x)
x–5
22. Consider this sequence.
3 3
3
3
3
,…
3, ___, ____, _____, ______, … , _____
10 100 1000 10000
10n –1
Show that the limit of the sum of the
10
terms in this sequence is ___.
3
23. Show that the limit of the sum of the
terms in this sequence does not exist.
1, –2, 3, –4, 5, … , n (– 1)n + 1, …
24. Determine the limit of each continued
fraction. What do you notice?
a)
1
3 +______________________
9
6 + __________________
25
_______________
6+
49
____________
6+
81
________
6+
121
6 + ____
b)
4
_______________________________
1
___________________________
1+
4
_______________________
3+
9
____________________
5+
16
________________
7+
25
9 + ____________
36
________
11 +
49
13 + ___
1.3 Limits • MHR
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